document_title: Kylo Ren

question_text: who did kylo ren kill first in the force awakens

text: Kylo Ren

text: Concept and creation[edit]

text: After Lucasfilm was acquired by The Walt Disney Company in 2012, the script for The Force Awakens was written by Lawrence Kasdan, J. J. Abrams and Michael Arndt, with no direct participation from Star Wars creator George Lucas. According to director Abrams, the First Order is inspired by ODESSA, a theoretical network of SS officers who fled to Argentina following World War II.[4]

text: Driver's casting in the film in an unnamed role was first announced on April 29, 2014.[5] Kylo Ren was first seen, but still not named, in the 88-second The Force Awakens teaser trailer released by Lucasfilm on November 28, 2014,[6][7] wielding a jagged red lightsaber with a crossguard.[8][9][10] The name Kylo Ren was revealed by Entertainment Weekly in a Lucasfilm-designed Topps-style trading card mockup in December 2014.[11][12] A May 2015 Vanity Fair photo shoot by Annie Leibovitz confirmed that Driver would be portraying Kylo.[13]

text: In an interview with Time, The Force Awakens costume designer Michael Kaplan said that Abrams requested that Ren's mask be designed to be memorable for a child. After numerous attempts to gain Abrams' approval, the final design was selected. Kaplan said:

text: I don't know if it was the kind of spaghetti type lines on it or what, but the next time J.J. came by that was what we presented to him and he loved it. Also the silver in those lines kind of reflects and changes color with the action. You know, if he's standing in front of fire you see that, so it almost brings you into the mask.[14]

text: Abrams told Entertainment Weekly in August 2015, "The movie explains the origins of the mask and where it's from, but the design was meant to be a nod to the Vader mask."[15]

text: According to other cast members, Driver is a method actor, which meant that he sometimes stayed in character on set as Ren and left his mask on between scenes.[16] Driver explained that his goal was "to forget you're in Star Wars and treat it like any other job that's filled with moments and problems", because from the perspective of the characters living within the film's universe, "Darth Vader is real".[16]

text: Ren's character arc shares similarities with that of the Star Wars expanded universe character Jacen Solo, the son of Han Solo and Princess Leia who threatens the galaxy as a fallen Jedi.[17][18][19][20] Additionally, critics have noted a resemblance between Ren's character design and that of Revan, the protagonist of Knights of the Old Republic.[21][22]

text: Character[edit]

text: Abrams told Empire in August 2015, "Kylo Ren is not a Sith. He works under Supreme Leader Snoke, who is a powerful figure on the Dark Side of the Force."[4] Abrams had previously told Entertainment Weekly that the character, "came to the name Kylo Ren when he joined a group called the Knights of Ren."[15] Robbie Collin of The Telegraph described Ren as "a hot-headed, radicalised Dark Side jihadi, whose red lightsaber splutters and crackles as violently as his temper".[23] Abrams noted, "The lightsaber is something that he built himself, and is as dangerous and as fierce and as ragged as the character."[15] The Telegraph also explains that Ren's wild and erratic temper and "angsty" instability make him dangerous.[24] Melissa Leon of The Daily Beast describes Ren's use of the Force as "formidable", citing his ability to stop a blaster shot mid-air, immobilize victims and probe their minds against their will.[25]

text: Kasdan told Entertainment Weekly in August 2015, "I've written four Star Wars movies now, and there's never been a character quite like the one that Adam plays. I think you're going to see something that's brand new to the saga," noting that the character is "full of emotion".[15] Abrams explained, "I think that what makes Ren so unique is that he isn't as fully formed as when we meet a character such as Darth Vader ... He is not your prototypical mustache-twirling bad guy. He is a little bit more complex than that."[15] Driver said in December 2015 that, despite the visual similarities to Darth Vader, Ren is "unlike any villain the franchise has seen before".[26] He explained:

text: I feel there's a recklessness about him that's maybe not normally associated with the Dark Side. You normally think of order, and structure, and full commitment and no hesitation ... he's just a little bit more unpolished. It's in his costume, in his lightsaber—how you kind of get the sense that it could just not work at any moment; that it could just blow up. That’s kind of like a big metaphor for him.[26]

text: Appearances[edit]

text: Feature films[edit]

text: The Force Awakens (2015)[edit]

text: Kylo Ren first appears in The Force Awakens as a commander in the First Order, a sinister regime that has risen from the remains of the old Galactic Empire. After arriving at Jakku to retrieve a map containing the coordinates where Luke Skywalker is to be found, Ren kills an old vicar known as Lor San Tekka, and captures Resistance pilot Poe Dameron, who has also been sent to recover the map by General Leia Organa. Ren soon learns that the pilot had entrusted his astromech droid, BB-8, with the map. Poe flees with the help of rogue stormtrooper Finn, who later finds BB-8, and the scavenger Rey. Finn, Rey, and BB-8 escape Jakku in the Millennium Falcon, and are soon intercepted by the ship's former owner, Han Solo, and his co-pilot Chewbacca.

text: It is revealed that Ren is the son of Han and Leia, and was once one of Luke's Jedi pupils. He was drawn to the dark side of the Force by the First Order's Supreme Leader Snoke. He then helped to destroy his uncle Luke's new Jedi Academy. However, Ren still feels the pull of the light side of the Force and seeks the strength to overcome this weakness from his grandfather Darth Vader, whose burnt helmet is in Ren's possession. Arriving at Maz Kanata's castle on Takodana, Ren captures Rey, who he senses has seen the map. While interrogating her with the Force, he realizes that she is strong with the Force herself, though unaware of it. Rey finds herself able to resist his powers and experience Ren's emotions, and confronts him over his fear that he will never be as powerful as his grandfather. She later uses the "Jedi mind trick" to compel her stormtrooper guard to let her escape. Han arrives at the First Order's battle station, Starkiller Base, as part of the Resistance's plan to destroy it.

text: After Han plants explosives to destroy the base, he confronts Ren—calling him by his real name, Ben—and implores him to abandon the dark side, warning him that Snoke will kill him once he accomplishes his goal to seize control of the galaxy. Despite being conflicted, Ren refuses and kills his father, enraging Chewbacca, Han's copilot, who fires, injuring Ren. As Finn and Rey flee the damaged base, a badly injured Ren follows and confronts them. Finn fights Ren with Luke's recovered lightsaber, but Ren overpowers and severely wounds him. Rey then takes up the lightsaber and, using the Force, begins to overcome Ren, striking him on the face with the lightsaber and leaving with a scar on the right side of his face. Before the duel is finished, they are separated by a seismic fissure created by the collapsing base. Rey and the others escape as Snoke orders General Hux to evacuate the base and bring Ren to him to complete his training.

text: The Last Jedi (2017)[edit]

text: Ren's inner conflict continues into The Last Jedi, specifically through his conversations with Rey, with whom he connects through the Force. Rey believes that his conflict will ultimately bring him back to the light side. Ren captures her and brings her before Snoke, who orders Ren to kill her. However, Ren uses the Force to ignite Luke's lightsaber at Snoke’s side and cut him in half, killing him. After that, Ren reveals to Rey his new goal to create a new order in the galaxy, separate from the legacies created by Snoke and Luke, and beseeches Rey to join him, while explaining to her about her true parentage, calling Rey a "nobody" to everyone else in the galaxy except for Ren. Though tempted, Rey refuses, realizing that Ren will not turn back to the light side. After she escapes, Ren frames her for Snoke's assassination, declares himself Supreme Leader of the First Order, and orders his forces to attack the Resistance base on Crait. Ren engages Luke in battle and appears to blow him up and cut him in half, but Luke remains standing; “Luke” is revealed to be simply a vision projected from the real Luke on Ahch-To, serving as a distraction which allows the Resistance to escape from the First Order. When Luke vanishes, the First Order storms the base, but the Resistance has already evacuated; Ren shares a final look with Rey through the Force before Rey closes the door to the Millennium Falcon and escapes with the Resistance.

text: Related works and promotional material[edit]

text: He is a playable character in the 2015 Force Awakens add-on to the Disney Infinity 3.0 video game, with an Infinity character figurine available separately.[27][28][29][30] He is also a character in the strategy video game Star Wars: Force Arena.[31]

text: Hasbro has released a 3 3⁄4-inch (9.5 cm) Kylo Ren action figure,[32] and a 6-inch (15 cm) figure in their Black Series line.[33] He is also featured in the Lego Star Wars playsets Kylo Ren's Command Shuttle (2015)[34][35] and Battle on Takodana (2016),[36][37] as well as a Lego Buildable Figure.[38] The Lego version of Kylo also appears in the 2016 short form animated series Lego Star Wars: The Resistance Rises,[39][40][41][42] and as a playable character in Lego Star Wars: The Force Awakens.[43]

text: In January 2016, Driver reprised the role for a Star Wars/Undercover Boss sketch on Saturday Night Live, with Kylo Ren disguising himself as a radar technician named "Matt" to determine what the Starkiller Base employees really think of him.[44]

text: Kylo Ren appears in Star Wars Battlefront II, voiced by Matthew Wood and Roger Craig Smith, masked and unmasked respectively. In the game, Kylo interrogates Del Meeko about Lor San Tekka's location (who possess the map to Luke Skywalker) using his Force abilities. When Del finally relents and reveals the map and Lor San Tekka's location, Kylo leaves him for Hask, Del's former comrade in Inferno Squad.[45]

text: Novels[edit]

text: Kylo Ren is a point of view character in the 2015 novelization of The Force Awakens by Alan Dean Foster.[46]

text: In the 2017 Chuck Wendig novel Star Wars: Aftermath: Empire's End, Ben Solo is stated as having been born on the planet Chandrila on the same day as a peace treaty is signed between the remnants of the Empire and the New Galactic Republic.[47] Kylo Ren is additionally mentioned in the 2017 novel Star Wars: Phasma.

text: Reception[edit]

text: Both the character of Kylo Ren and Driver's performance have been positively received. Many complimented Ren's conflicted nature and depth, as well as his costume design, and noted there were many places the character could be taken in future installments.[48][49][50] Peter Bradshaw of The Guardian praised the character and the actor alike, saying, "He is gorgeously cruel, spiteful and capricious – and unlike the Vader of old, he is given to petulant temper tantrums, with his lightsaber drawn."[51] Terri Schwartz of IGN also called Driver's performance "spectacular", noting that "his performance adds great depth to a character who could have come off as one-dimensional, and the implications of his arc leave a viewer with plenty to think about after they leave the theater." [52] Collin wrote, "To describe Kylo Ren as this film's Vader would be accurate in a sense ... But it would also be to undersell the deep ingenuity with which this astonishing character has been crafted by Abrams, Kasdan and Arndt, and also the wells of emotional tumult Driver invests in him."[23] Comparing the character to the one-note Vader of the 1977 film, Leon calls Ren "a living battleground between darkness and light, making him a far more resonant and familiar portrayal of that struggle than we've ever seen in Star Wars ... [which] makes him a far more interesting villain."[25] Abrams told Entertainment Weekly, "it was a great joy to work with Adam Driver on this role, because he threw himself into it in a deep and remarkable way."[15] Todd McCarthy of The Hollywood Reporter noted, "Ren is given a pronounced inferiority complex, a clever bad guy twist that could be taken to interesting places both in the writing and performance."[53] Peter Travers of Rolling Stone wrote, "The bald-faced attempt to clone Vader, one of the greatest badasses in film history, is clankingly obvious, but Driver, masked and unmasked, gives him hypnotic and haunting contours."[54] Kyle Buchanan of Vulture.com was underwhelmed by the reveal of Driver under the mask.[55] Leon, however, argued:

text: But that face—that of a normal, vulnerable young man—is the most subversively terrifying thing about J. J. Abrams' reimagining of A New Hope. Rather than pure evil, Ren is something far more familiar: He is human. Just like the real-life young men with minds clouded by fear, hate, and anger who commit unspeakable acts in our world every day ... all the visual cues that leave the character open to criticisms of not being "evil" enough—are all signs of Ren's struggle between the Dark Side and the Light.[25]

text: Driver won a 2016 Saturn Award for Best Supporting Actor for his portrayal.[56] In January 2018, Kylo was voted seventh greatest movie villain of all time by the readers of Empire.[57]

text: Family tree[edit]

text:
hide

vteSkywalker family tree

   AikaLars[a]   ClieggLars[b] ShmiSkywalker[c]  JobalNaberrie[d]   RuweeNaberrie[d]

BeruWhitesun[e]   OwenLars[f]    AnakinSkywalker[g][h]     PadméAmidala[i]   BailOrgana[j][k] BrehaOrgana[l]

                 LukeSkywalker[m][n] LeiaOrgana[o]   HanSolo[p]

                        BenSolo[p][q]

References:

Jump up ^ Hidalgo, Pablo (2016). Star Wars Character Encyclopedia: Updated and Expanded.

Jump up ^ "Cliegg Lars". StarWars.com. Retrieved December 3, 2015.

Jump up ^ "Shmi Skywalker Lars". StarWars.com. Retrieved December 3, 2015.

^ Jump up to: a b Attack of the Clones (2002) and Revenge of the Sith (2005)

Jump up ^ "Beru Lars". StarWars.com. Retrieved December 3, 2015.

Jump up ^ "Owen Lars". StarWars.com. Retrieved December 3, 2015.

Jump up ^ "Anakin Skywalker". StarWars.com. Retrieved December 4, 2015.

Jump up ^ "Darth Vader". StarWars.com. Retrieved December 4, 2015.

Jump up ^ "Padmé Amidala". StarWars.com. Retrieved December 4, 2015.

Jump up ^ Adopted father of Leia Organa, as established in Revenge of the Sith (2005).

Jump up ^ "Bail Organa". StarWars.com. Retrieved December 3, 2015.

Jump up ^ Adopted mother of Leia Organa, as established in Revenge of the Sith (2005).

Jump up ^ "Luke Skywalker". StarWars.com. Retrieved December 4, 2015.

Jump up ^ In the alternate Star Wars expanded universe (Legends), Luke is married to Mara Jade and has a son, Ben Skywalker.

Jump up ^ "Princess Leia Organa". StarWars.com. Retrieved December 4, 2015.

^ Jump up to: a b The Force Awakens. 2015.

Jump up ^ In the alternate Star Wars expanded universe (Legends) continuity, Han and Leia have three children: Jaina, Jacen and Anakin Solo.


document_title: List of To Kill a Mockingbird characters

question_text: what is tom's full name in to kill a mockingbird

question_tokens: what is tom 's full name in to kill a mockingbird

root

text: Primary characters[edit]

text: Atticus Finch[edit]

text: Atticus Finch is the middle-aged father of Jem and Scout Finch. He is a lawyer and was once known as "the deadliest shot in Maycomb County". Although he was a good shot, he does not like to mention the fact as he does not like the thought of having an advantage over people. He appears to support racial equality and was appointed to represent Tom Robinson, a black man who has been accused of raping a young white woman, Mayella Ewell. The town disapproves of him defending Tom especially when he makes clear his intent to defend Tom Robinson to the best of his abilities. He is portrayed by Gregory Peck in the film adaptation of To Kill a Mockingbird.

tag: p

text: "Scout" Finch[edit]

text: Jean Louise "Scout" Finch is the narrator and To Kill a Mockingbird is told by an adult Scout who often comments on how she could not understand something at the time but now can appreciate it. She gets in trouble with her teacher Miss Caroline because Miss Caroline expects Scout to learn reading and writing her way. She is a tomboy and spends the most of her time with her brother Jem and best friend Dill. To Jem’s advice to pretend to be a lady and start sewing or something, she answers ‘Hell no.’ The rare hints the narrator gives us about her grown-up life reveal that she ultimately hasn’t changed herself for others.[2]

text: She matures from age 6 to age 9 as the novel progresses but remains naive and idealistic, despite an increased understanding of human nature and racism in her town. At the beginning of the book, Scout is confused by some of the words and names she hears people directing towards her father, such as "nigger-lover". Being only six, Scout does not know how to handle such situations so she tries to resolve her problems by fighting, or by talking to Atticus about what she has heard. By the end of the book, Scout realizes that racism does exist and comes to terms with its presence in her town. Scout also learns how to deal with others, including the Finch family housekeeper, Calpurnia, and her aunt, Alexandra. Scout is the only one of the novel's primary three children (Dill, Jem, and herself) to see and speak to Boo Radley during the course of the novel and realize that he is harmless, despite her initial fear of him. She also stops a mob that is trying to lynch Tom Robinson by talking to the mob leader, Mr. Cunningham, about inviting his son, Walter, over for dinner. Mr. Cunningham then tells the other mob members to get back in their cars and leave them alone. The members listen, and Scout unintentionally saves Tom Robinson's life. She is portrayed by Mary Badham in the film adaption of To Kill a Mockingbird.

text: Notable quote: "Until I feared I would lose it, I never loved to read. One does not love breathing." [3]

text: "Jem" Finch[edit]

text: Jeremy Atticus "Jem" Finch is Atticus' son and Scout's older brother by four years. Jem matures greatly throughout the course of the novel, much more affected by events than Scout seems to be. Being four years Scout's senior, Jem is seen to have a greater understanding of - and therefore greater difficulty in navigating - the obstacles thrown their way. Jem is seen explaining many things to Scout throughout the novel. Bob Ewell breaks Jem's arm, subsequently resulting in it being shorter than it had been, in an attempt to protect his sister. He is portrayed by Phillip Alford in the film adaptation of To Kill a Mockingbird.

text: Dill Harris[edit]

text: Charles Baker "Dill" Harris is a short, smart boy who visits Maycomb every summer from Meridian, Mississippi, and stays with his Aunt Rachel (Aunt Stephanie in the film). Dill is the best friend of both Jem and Scout, and his goal throughout the novel is to get Boo Radley to come out of his house. The children concoct many plans to lure Boo Radley out of his house for a few summers until Atticus tells them to stop. In chapter 5 of the novel, Dill promises to marry Scout and they become "engaged". One night Dill runs away from his home in Meridian, arriving in Maycomb County where he hides under Scout's bed. When she finds Dill, he tells both Scout and Atticus that he was chained to a wall in his father's basement; later, he confesses he actually ran away because he felt he was being replaced by his stepfather. Before Dill returned to Meridian after the summer, he went swimming with Jem at the Barker's Eddy creek. Scout, unfortunately, was unable to participate, because both boys were swimming naked.

text: Unlike Scout and Jem, Dill lacks the security of family support. He is unwanted and unloved by his mother and stepfather: "They do get on a lot better without me, I cannot help them any." As Francis (a cousin of Jem and Scout) says, "He hasn't got a home, he just gets passed around from relative to relative." Dill maintains he has no father; he doesn't know where his father might live, or if he will ever come back. He is played by John Megna in the film adaption of To Kill a Mockingbird.

text: This character is believed to be based on author Truman Capote, a childhood friend of Harper Lee.[citation needed]

text: Calpurnia[edit]

text: Calpurnia, nicknamed Cal, is the Finch family's African-American housekeeper, whom the children love and Atticus deeply respects (he remarks in her defense that she "never indulged [the children] like most colored nurses"). She is highly regarded by Atticus. She is an important figure in Scout's life, providing discipline, instruction, and love. She also fills the maternal role for the children after their mother's death. Calpurnia is a mother herself and raised her son, Zeebo, to adulthood. Calpurnia is one of the few black characters in the novel who is able to read and write, and it is she who taught Scout to write. She learned how to read from Miss Maudie's aunt, Miss Buford, who taught her how to read out of Blackstone's Commentaries, a book given to her. Aunt Alexandra despised Calpurnia because Alexandra believed that Calpurnia was not a "maternal figure" for Jem and Scout, especially for Scout.

text: Calpurnia is a member of the First Purchase M.E. African Church in Maycomb. While Scout always hears her speak proper English, she is surprised to learn that Calpurnia does not do so at church, especially with the uneducated members of the congregation.

text: While everyone in the novel is filtered through Scout’s perception, Calpurnia in particular appears for a long time more as Scout’s idea of her than as a real person. At the beginning of the novel, Scout appears to think of Calpurnia as the wicked stepmother to Scout’s own Cinderella. However, towards the end of the book, Scout views Calpurnia as someone she can look up to, and realizes Calpurnia has only protected her over the years. She is played by Estelle Evans in the film.

text: Arthur “Boo” Radley[edit]

text: Arthur "Boo" Radley is a recluse in To Kill a Mockingbird.

text: Maycomb children believe he is a horrible person, due to the rumors spread about him and a trial he underwent as a teenager. It is implied during the story that Boo is a very lonely man who attempts to reach out to Jem and Scout for love and friendship, such as leaving them small gifts and figures in a tree knothole. Scout finally meets him at the very end of the book, when he saves the children's lives from Bob Ewell. Scout describes him as being sickly white, with a thin mouth, thin and feathery hair, and grey eyes, almost as if he were blind. During the same night, when Boo whispers to Scout to walk him back to the Radley house, Scout takes a moment to picture what it would be like to be Boo Radley. While standing on his porch, she realizes his "exile" inside his house is really not that lonely.

text: Boo Radley's heroics in protecting the children from Bob Ewell are covered up by Atticus, Sheriff Tate, and Scout. This can be read as a wise refusal of fame. As Tate notes, if word got out that Boo killed Ewell, Boo would be inundated with gifts and visits, calamitous for him due to his reclusive personality. The precocious Scout recognizes the danger: renown would "kill the mockingbird". Boo Radley is a ghost who haunts the book yet manifests himself at just the right moments in just the right way. He is, arguably, the most potent character in the whole book and as such, inspires the other key characters to save him when he needs saving.

text: After the Tom Robinson trial, Jem and Scout have a different understanding of Boo Radley. “Scout, I think I'm beginning to understand something. I think I'm beginning to understand why Boo Radley's stayed shut up in the house all this time… it's because he wants to stay inside." (23.117) Having seen a sample of the horrible things their fellow townspeople can do, choosing to stay out of the mess of humanity doesn’t seem like such a strange choice.[4]

text: When Boo finally does come out, he has a good reason: Bob Ewell is trying to murder the Finch children. No one sees what happens in the scuffle, but at the end of it, Ewell is dead and Boo carries an unconscious Jem to the Finch house. Finally faced with Boo, Scout doesn’t recognize him at first, but suddenly realizes who he is. Boo Radley is played by Robert Duvall in the film.

text: Judge John Taylor[edit]

text: Judge John Taylor is a white-haired old man with a reputation for running his court in an informal fashion and an enjoyment of singing and dipping tobacco. He is unimportant to the children until he presides over the Tom Robinson trial, in which he shows great distaste for the Ewells and great respect for Atticus. Because of the judge's sympathies for Tom, Bob Ewell breaks into the judge's house while the judge's wife is at church. After the trial, Miss Maudie points out to the children that the judge had tried to help Tom by appointing Atticus to the case instead of Maxwell Green, the new, untried lawyer who usually received court-appointed cases. Judge Taylor knew that Atticus was the only man who would stand a chance at acquitting Tom, or at least would be able to keep the jury thinking for more than just a few minutes. By doing this, Judge Taylor was not giving in or supporting racism. He is portrayed in the film by Paul Fix.

text: Miss Maudie Atkinson[edit]

text: Miss Maude "Maudie" Atkinson lives across the street from the Finch family. She had known the Finches for many years, having been brought up on the Buford place, which was near the Finches' ancestral home, Finch's Landing. She is described as a woman of about 50 who enjoys baking and gardening; her cakes are especially held in high regard. However, she is frequently harassed by devout "Foot-Washing Baptists", who tell her that her enjoyment of gardening is a sin. The Foot-Washing Baptists also believe that women are a sin as well. Miss Maudie befriends Scout and Jem and tells them stories about Atticus as a boy. It is strongly implied that she and Atticus have a more than platonic relationship. Also, she is one of the few adults that Jem and Scout hold in high regard and respect. She does not act condescendingly towards them, even though they are young children. During the course of the novel, her house burns down; however, she shows remarkable courage throughout this (even saying that she wanted to burn it down herself to make more room for her flowers). She is not prejudiced, though she talks caustically to Miss Stephanie Crawford, unlike many of her Southern neighbors, and teaches Scout important lessons about racism and human nature. It is important to note that Miss Maudie fully explains that "it is a sin to kill a mockingbird", whereas Atticus Finch initially brings up the subject but doesn't go into depth. When Jem gets older, and doesn't want to be bothered by Scout, Miss Maudie keeps her from getting angry. Maudie is played by Rosemary Murphy in the film. She is a very kind and sweet innocent friendly neighbor.

text: Robert E. Lee "Bob" Ewell[edit]

text: Robert E. Lee "Bob" Ewell is the main antagonist of To Kill a Mockingbird. He has a daughter named Mayella and a younger son named Burris, as well as six other unnamed children. He is an alcoholic, poaching game to feed his family because he spends whatever money they legally gain via government "relief checks" on alcohol. It is implied, and evidence suggests, that he was the one who abused his daughter Mayella, not Tom Robinson (the African American man accused of doing so). Although most everybody in town knows that the Ewells are a disgrace and not to be trusted, it is made clear that Tom Robinson was convicted because he is a Negro whose accuser is white. Upon hearing of Tom's death, Bob is absolutely gleeful, gloating about his success. After being humiliated at the trial, however, he goes on a quest for revenge, becoming increasingly violent. He begins by spitting in Atticus' face, followed by a failed attempt to break into the home of Judge Taylor, then finally menacing Helen, the poor widow of Tom Robinson. Ewell later attempts to murder Jem and Scout Finch with a knife to complete his revenge. Boo Radley saves Jem and Scout and it is believed that he kills Ewell with the knife. Heck Tate, the sheriff, puts in the official report that Bob Ewell fell on his own knife and died after lying under a tree for 45 minutes. Ewell is played by James Anderson in the film.

text: Mayella Ewell[edit]

text: Mayella Violet Ewell, 19, is the oldest of the eight Ewell children. Before the trial, Mayella is noted for growing red geraniums outside her otherwise dirty home in order to bring some beauty into her life. Due to her family's living situation, Mayella has no opportunity for human contact or love. She eventually gets so desperate that she attempts to seduce a black man, Tom Robinson, by saving up nickels to send her siblings to go get ice cream so that Mayella can be alone with Tom. Her father sees this through a window and in punishment he beats her. Ewell then finds the sheriff, Heck Tate, and tells him that his daughter has been raped and beaten by Tom. At the trial, Atticus points out that only the right side of Mayella's face is injured, suggesting a left-handed assailant; Tom's left arm is mangled and useless, but Bob Ewell is left-handed. When Atticus asks her if she has any friends, she becomes confused because she does not know what a friend is. During her testimony she is confused by Atticus' polite speech and thinks that his use of "Miss Mayella" is meant to mock her. She testifies against Tom Robinson. Mayella is played by Collin Wilcox in the film.

text: Tom Robinson[edit]

text: Thomas "Tom" Robinson is an African-American who is put on trial for the rape of a white woman, Mayella Ewell. Atticus is assigned to defend him, and stands up to a lynch mob intent on exacting their own justice against him before the trial begins. Tom's left arm is crippled and useless, the result of an accident with a cotton gin when he was a child. Atticus uses this fact as the cornerstone of his defense strategy, pointing out that the nature of Mayella's facial injuries strongly suggest a left-handed assailant. Tom testifies that he had frequently helped Mayella with household chores because he felt sorry for her and the family's difficult life - a statement that shocks the all-white, male jury. Despite Atticus' skilled defense, the jury's racial prejudices lead them to find Tom guilty. Atticus plans to appeal the verdict, but before he can do so, Tom is shot and killed while trying to escape the prison where he is being held. Tom Robinson is played by Brock Peters.

text: Aunt Alexandra[edit]

text: Alexandra Hancock (née Finch) is Atticus' and Jack's sister, married to James "Uncle Jimmy" Hancock. She has a son named Henry and a very spoiled grandson named Francis. Around the middle of the book, Aunt Alexandra decides to leave her husband at Finch's Landing, the Finch family homestead to come stay with the Finches. Aunt Alexandra doesn't consider the black Calpurnia to be a very good motherly figure for Jem and Scout; she disapproves of Scout being a tomboy and wants to make Scout into a southern belle (encouraging her to act more 'lady like').[5] This is the cause of many conflicts between Scout and Alexandra throughout the course of the novel. However, as the trial progresses, Scout comes to see how much her aunt cares for her father and what a strong woman she is. This is especially evidenced by a tea party when Scout is horrified by the racism displayed, and her aunt and Miss Maudie help her deal with her feelings. By the end of the book, it's clear that Alexandra cares very much for her niece and nephew, though she and Scout will probably never really get along.

text: Minor characters[edit]

text: Jack Finch[edit]

text: John Hale "Jack" Finch is Atticus' and Alexandra's younger brother. (He is about 40, which is 10 years younger than Atticus.) Jack smells like alcohol and something sweet, and is said that he and Alexandra have similar features. Jack is a childless doctor who can always make Scout and Jem laugh, and they adore him. He and Miss Maudie are close to the same age; he frequently teases her with marriage proposals, which she always declines. Jack also has a pet cat named Rose Aylmer, who is mentioned during the Christmas visit.

text: Francis Hancock[edit]

text: Francis Hancock is the spoiled grandson of Aunt Alexandra. (The son of her son, Henry Hancock.) Every Christmas, Henry and his wife drop Francis at Finch's Landing, which is the only time Scout and Jem see him. Francis lives in Mobile, Alabama, and is a bit of a tattle-tale. He gets along well with Jem, but often spars with Scout. One Christmas, Francis calls Atticus a "nigger-lover," as well as insisting that he was ruining the family and the likes, which infuriates Scout and causes them to get into a fight. Francis lies about his role in it, telling Uncle Jack that Scout started it by calling him a "whore lady", and Jack therefore punishes Scout. However, she explains the full story and charitably persuades her uncle not to punish Francis about it, but to let Atticus think they had been fighting about something else (although Atticus later discovers the truth).

text: Mrs. Henry Lafayette Dubose[edit]

text: Mrs. Henry Lafayette Dubose is an elderly woman who lives near the Finches. She is hated by the children, who run by her house to avoid her. Scout describes Mrs. Dubose as "plain hell." A virulent racist, she calls Atticus a "nigger-lover" to his children's faces, and Jem flies into a rage and ravages Mrs. Dubose's camellia bushes. As a punishment, Jem is required to read to Mrs. Dubose each day for a month. As Jem reads, she experiences a fit of drooling and twitching and does not seem to pay any attention to the words. When an alarm rings, Jem is allowed to leave for the day. She extends the punishment for one extra week and dies shortly after letting Jem go for the last time. Atticus informs him that Mrs. Dubose was terminally ill and had become addicted to morphine. By reading to her, Jem had distracted her so that she could break the addiction. In thanks, she leaves him a candy box with a camellia flower in it; Jem burns the box in anger, but is later seen by Scout admiring the flower. Atticus tells Jem that Mrs. Dubose was the bravest person he ever knew, and she was trying to teach Jem the importance of bravery and true courage to endure anything when the situation is hopeless, as in her morphine addiction.

text: Mr. Heck Tate[edit]

text: Mr. Heck Tate is a friend of Atticus and also the sheriff of Maycomb County. He believes in protecting the innocent although he doesn't usually show it. At the end of the book, the Atticus and Heck argue over whether Jem or Boo Radley should be held responsible for the death of Bob Ewell. Heck eventually persuades Atticus to accept the theory that Ewell accidentally fell on his own knife, thus saving the harmless, reclusive Boo from the public exposure of a criminal trial.

text: Mr. Braxton Underwood[edit]

text: Mr. Braxton Bragg Underwood is a news reporter and a friend of Atticus. He owns and also publishes The Maycomb Tribune. Being a racist, he disagrees with Atticus on principle. He also has a strong belief in justice, as exemplified when he defends Atticus from the Cunningham mob by having his double barrel shotgun loaded and ready to shoot them. He also demonstrates some humanity when he publishes a scathing editorial comparing the killing of Tom Robinson (a cripple) to "the senseless slaughter of songbirds by hunters and children".

text: Mr. Horace Gilmer[edit]

text: Mr. Horace Gilmer is a lawyer from Abbottsville, and is the prosecuting attorney in the Tom Robinson case. Mr. Gilmer is between the ages of forty and sixty. Mr. Gilmer has a slight cast with one eye, which he uses to his advantage in trial. Mr. Gilmer appeared to be racist in his harsh cross-examination of Tom Robinson, but it is hinted at that he is in fact going easy on Tom. He and Atticus are not rivals and talk to each other during recesses during the case.

text: Dr. Scott Reynolds[edit]

text: Dr. Reynolds is the Maycomb doctor. He is well known to Scout and Jem. Scout says that he "had brought Jem and me into the world, had led us through every childhood disease known to man including the time Jem fell out of the tree house, and he had never lost our friendship. Dr. Reynolds said that if we were boil-prone things would have been different..." (ch. 28). He inspects Jem's broken arm and Scout's minor bruises after the attack from Bob Ewell under the tree.

text: Dolphus Raymond[edit]

text: Dolphus Raymond is a white landowner who is jaded by the hypocrisy of the white society and prefers to live among black folks. In fact, he has children with a black woman. Dolphus pretends he is an alcoholic so that the people of Maycomb will have an excuse for his behavior, but in fact he only drinks Coca Cola out of a paper bag to try to hide it. When Dill and Scout discover that he is not a drunk, they are amazed. He shows Scout how sometimes you can pretend to be someone you're not so people will be more understanding of you.

text: Link Deas[edit]

text: Link Deas owns cotton fields and a store in Maycomb who employs Tom and later Helen because she does not get accepted by any other employers in the county due to Tom Robinson's legal troubles. He announces to the court (in defense of Tom) at one point in the trial that he hadn't “had a speck o' trouble outta him” in the eight years Tom had been working for him, and gets sent out by Judge John Taylor for doing so. When Bob Ewell starts threatening Helen after the trial, Mr. Deas fiercely defends her and threatens to have Mr. Ewell arrested if he keeps bothering her. He is on Tom Robinson's side during the trial and remains loyal to the family afterward.

text: Miss Caroline Fisher[edit]

text: Miss Caroline Fisher is Scout's first grade teacher and is new to Maycomb, Alabama and its ways. She attempts to teach the first grade class using a new system which she learned from taking certain college courses (Jem mistakenly refers to it as the "Dewey Decimal System", which is really how library books are organized.). She is upset by Scout's advanced reading capabilities and believes that Scout is receiving lessons from Atticus. She feels as though Scout is trying to outsmart and mock her. In an effort to standardize the class, she forbids Scout from reading with her father. Atticus asks Scout to step into Miss Caroline's skin. However, he continues to allow Scout to read with him at night so long as she continues to go to school. Miss Caroline has good intentions but proves quite incompetent as a teacher. When Scout tells Miss Fisher that she shamed a student (Walter Cunningham, Jr.) by giving him lunch money, she raps Scout's palms with a ruler (a punishment unheard of in Maycomb). She is also very sensitive and gets emotionally hurt quite easily, as seen when she cries after Burris Ewell yells at her, "Report and be damned to ye! Ain't no snot-nosed slut of a schoolteacher ever born c'n make me do nothin'! You ain't makin' me go nowhere, missus. You just remember that, you ain't makin' me go nowhere!" After the Burris Ewell incident, Miss Caroline is seldom seen and soon forgotten.

text: Reverend Sykes[edit]

text: Reverend Sykes is the reverend of the First Purchase M.E. African Church in Maycomb County. This is the church Tom Robinson attended. Reverend Sykes forces the congregation to donate 10 dollars for Tom Robinson's family since at the time, Tom's wife, Helen, was having trouble finding work. During the trial, when the courtroom was too packed for the children to find seats, Reverend Sykes lets the kids sit with him up in the colored balcony and even saves their seats for them.

text: Miss Stephanie Crawford[edit]

text: Stephanie Crawford is the neighborhood gossip who claimed that she saw Boo Radley from her bedroom standing outside of her cleaned window one night. Crawford is one of the first on the scene after a loud gunshot is heard behind the Radley house. Because she is the neighborhood gossip, it is unwise to think of anything that she says as true, because most of the time it is not true at all. She is a friend of Alexandra Hancock. She lets Miss Maudie live with her when Miss Maudie's house burns down, supposedly in order to steal Miss Maudie's Lane cake recipe. She is thrilled to pass on gossip to the kids about Boo Radley. She claimed to have witnessed Bob Ewell's threatening Atticus at the Post Office corner as she was returning from the local Jitney Jungle grocery store.

text: In the film, she takes the place of Rachel Haverford and is now Dill's aunt.

text: Miss Rachel Haverford[edit]

text: Miss Rachel Haverford is Dill's aunt and the Finches' next door neighbor. She drank neat whiskey heavily after seeing a rattlesnake coiled in her closet, on her washing, when she hung her negligee up. Even though she can be very hard to deal with, she truly does love her nephew. Her family name, in the legends of Maycomb County, is synonymous with jackass. She is also a Southern Belle.

text: In the film, she is not a character and Miss Stephanie takes her place as Dill's aunt.

text: Helen Robinson[edit]

text: Helen Robinson is the wife of Tom Robinson. She is spoken about a few times. She has three children. Employed by Link Deas following the death of her husband, she is repeatedly harassed by Bob Ewell when traveling to work. Upon learning of this, Deas threatens Ewell, forcing him to stop. She is an example of how one person's actions can have an effect on a lot of people and she elucidates the hardships that surround the Tom Robinson case.

text: Nathan Radley[edit]

text: Nathan Radley is the brother of Arthur "Boo" Radley and another difficult character to understand in To Kill a Mockingbird. When the children try to catch a view of "Boo" late one night through a window, he shoots over their heads with a shotgun (albeit thinking he was aiming at a black person). Nathan also cements up the knothole in which Arthur leaves little gifts for the children. On the other hand, he helps Miss Maudie by saving some of her belongings when her house is on fire. He is more present than his brother, but equally mysterious.

text: Jessie[edit]

text: Jessie is Mrs. Dubose's black nurse. She is the woman who shoos the children out when Mrs. Dubose has her fits, and she seems to care enormously for Mrs. Dubose. When Jem is forced to stay reading to Mrs. Dubose, Jessie kindly leads Jem and Scout to the door when Mrs. Dubose's alarm goes off. The rumors about Mrs. Dubose concealing a gun about her person involves Jessie; the book says "and even if Mrs. Dubose missed, her girl Jessie wouldn't".

text: Burris Ewell[edit]

text: Burris Ewell is a son of Bob Ewell and a younger sibling of Mayella Ewell as well as the first antagonist of the novel. Burris is described as being chiefly antagonistic of Little Chuck Little and his teacher Miss Caroline Fisher. He comes to the first day of school, but departs just as everyone else in his family does. Burris is also poor. He has live lice in his hair. Burris also scared his teacher Caroline Fisher. He behaves rudely when she tells him to go home, wash his hair, and come back clean the next day. He refuses, and a student explains to Miss Caroline that Ewell children don't attend school. All they do is show up for the first day, get marked down on the register, and then they miss the entire school year until the first day of the next year. His famous quote was, "Report and be damned to ye! Ain't no snot-nosed slut of a schoolteacher ever born c'n make me do nothin'! You ain't makin' me go nowhere, missus. You just remember that, you ain't makin' me go nowhere!" As of Scout's first year of school (the first grade), Burris has repeated the first grade three times. Burris is also like his father and is very belligerent.

text: Lula[edit]

text: Lula is an African-American woman with a dislike for white people. She doesn't like the idea of Calpurnia bringing Atticus Finch's children, Jem and Scout, with her to church and tells her so but is overruled by the other congregants. According to James Zeebo, Calpurnia's son, Lula's said to be, "a troublemaker from way back, with fancy ideas and haughty ways." She's threatened with being "churched" (subjected to church discipline) by Reverend Sykes.

text: Mrs. Grace Merriweather[edit]

text: Mrs. Grace Merriweather is the producer of the play in which Scout plays as a ham. She tells Everett that "the ladies of the South Maycomb Alabama Methodist Episcopal Church are behind him one hundred percent." She is mostly known for her devotion to the church and is widely held as the most devout lady in Maycomb; however, like many of her peers, she is very hypocritical and loves to gossip with all the other women.

text: Walter Cunningham, Jr.[edit]

text: Walter Cunningham, Jr. is a child who is almost as old as Jem but is in Scout's class. He lives on a farm. He is too poor to even pay off a 25-cent debt because the Great Depression hit his poor family hard. He doesn't take money because his family can't pay people back in cash. His father paid Atticus for his service for something a while back with some goods. Walter is invited over to the Finches' house once, after engaging in a fight with Scout, where he covers up all of his dinner with molasses, much to Scout's vocal dismay. This teaches Scout a lesson in humility and compassion.

text: Walter Cunningham, Sr.[edit]

text: Walter Cunningham, Sr. is Walter Cunningham Jr.'s father. He appears only twice, once at the beginning of the story when he has to pay off the debt to Atticus (Walter Cunningham Sr. was his client) by giving him firewood, vegetables and other supplies. He also leads the mob that comes to lynch Tom Robinson the night before the trial. Only when Scout talks to him about his son and how much he owes to Atticus does he reconsider and call off the mob. Scout innocently shames him because Scout reminds him of all the things that Atticus has done for him and for Maycomb County. After the verdict is given in the trial, Atticus tells Jem that one of the Cunninghams had changed his thoughts about Tom and pleaded that Tom was not guilty to the jury.

text: Little Chuck Little[edit]

text: Little Chuck Little is a student in Scout's first grade class who has the mindset of an adult. His real name is Charles. He is depicted as chiefly antagonistic of Burris Ewell. He is presented in the novel when Miss Caroline is frightened by Burris' lice. He warned Miss Caroline that if Burris wasn't released from class, he might try something that would put their classmates at risk. When Burris starts advancing on Little Chuck after his warning/veiled insult, Little Chuck's hand moved to his pocket (implying that he was going to pull out a knife) while saying, "Watch your step, Burris. I'd soon's kill you as look at you. Now go home." Scared by Little Chuck's bravery and his implied knife, Burris retreats. From this we see, through the narrative view of Scout, his gentlemanly attitude and how it calms Miss Caroline down. Little Chuck may be even more intelligent than originally meets the eye, as he easily could have been bluffing about the aforementioned implied knife to scare Burris into retreating.

text: Mr. Dick Avery[edit]

text: Mr. Dick Avery is an overweight neighbor who tells Jem and Scout that dramatic changes in the weather are caused by disobedient and misbehaving children. After it snows, they build a snowman that resembles him. Mr. Avery can also be seen in the story pushing a mattress out of a window when Miss Maudie's house catches fire. The kids, including Jem and Scout, always waited for him to do something interesting. For instance, Jem claims that one night he urinated from his front porch in an impressive arc.

text: Miss Gates[edit]

text: Miss Gates is a third grade teacher at Scout's school who insists that America isn't prejudiced like Hitler's Germany. Despite this, Scout has heard her say that the blacks need to be taught a lesson after Tom's trial.

text: Eula May[edit]

text: Eula May is Maycomb's most prominent telephone operator. She sends out public announcements, invitations, and activates the fire alarm. She announced the closing of schools when it snowed and announced the rabid dog that entered Maycomb. Also, Eula May knows everybody in the town because of her unique job.

text: Cecil Jacobs[edit]

text: Cecil Jacobs teases Scout and Jem at school. Scout almost gets into a fight with Cecil over the trial of Tom Robinson. Scout beats up Cecil Jacobs because he says Atticus is a "Nigger Lover." He gives a current event on Adolf Hitler and later frightens Scout and Jem on their way to the Halloween pageant. He and Scout then pair up at the carnival. At the Halloween pageant afterwards, Cecil was a cow. He hints that black people are not as good as white people while talking about Hitler during current events. He also tends to take jokes too far. However importantly he shows how prejudice is passed on from parent to child.

text: Tim Johnson[edit]

text: Tim Johnson is a dog belonging to Harry Johnson (a character in the book who is mentioned once but is never seen). He is infected by rabies in chapter 10 and goes mad, putting everyone in the town at risk. Atticus is forced to shoot Tim Johnson before he reaches the Radley House or attacks anyone. When Atticus shoots the dog, his excellent marksmanship is revealed to Scout and Jem (his nickname used to be One-Shot Finch). The dog's body is collected by Zeebo.

text: Simon Finch[edit]

text: Simon Finch is the founder of Finch's Landing. He is referred to in the first chapter of the book, being a direct ancestor of Atticus. He is a Cornish Methodist, and emigrated from England to avoid religious persecution, landing in Philadelphia before settling in Alabama. He was married, with one son, eight daughters. He is also an apothecary.

text: Maxwell Green[edit]

text: Maxwell Green is the new lawyer in town. He is normally the judicially-assigned defense attorney but Judge Taylor assigned Tom Robinson's case to Atticus to give Tom Robinson a better chance.

text: Mr. X Billups[edit]

text: Mr. X Billups who is seen only once in the book, going to the trial, is described as a "funny man". X is his name, and not his initial. He was asked repeated times what his name was until he signed it. X was the name he had been given when he was born because his parents marked his birth certificate with an X instead of a name.

text: The Barber Sisters (Miss "Tutti" and Miss "Frutti")[edit]

text: The Barber Sisters (Miss Sarah, nicknamed "Tutti" and Miss Frances, nicknamed "Frutti") are maiden sisters who live in the only house in Maycomb with a cellar. They were originally from Clanton, Alabama; and are rumored to be Republicans. Besides their Yankee ways, both sisters are deaf (Tutti completely deaf; Frutti uses an ear trumpet), and had a Halloween prank pulled on them by some "wicked" schoolchildren (Scout claims she was not included) who put all of their furniture in their cellar.

text: Mrs. Gertrude Farrow[edit]

text: Mrs. Farrow is a lady in the missionary society who visits the Finch house occasionally.

text: Mr. Conner[edit]

text: Mr. Conner is mentioned early on in the book. He was locked in an outhouse by "Boo" Radley and his friends. After taking the teenagers to court, Mr. Conner accused them of "disorderly conduct, disturbing the peace, assault and battery, and using abusive and profane language in the presence and hearing of a female." He added the last charge after claiming that the teens had "cussed so loud he was sure every lady in Maycomb heard them."

text: Zeebo[edit]

text: Zeebo is Calpurnia's eldest son. He is one of just four people in First Purchase Church who can read, so he is the vocal leader, leading hymns in the Negro First Purchase Church by "lining," reading a line of verse and having the congregation repeat it. Calpurnia taught Zeebo how to read. He is also the garbage man of Maycomb, and took away the dead rabid dog, Tim Johnson. When a fellow church member, a woman named Lula, tries to make the children feel bad for attending church with his mother, Zeebo welcomes Scout and Jem with open arms. He is kind but very quiet.


document_title: Limit of a function

question_text: when does a function not have a limit

question_tokens: when does a function not have a limit

text: Limit of a function

text: History[edit]

text:
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of

y
=
f
(
x
)

{\displaystyle y=f(x)}

 by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).

text: The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).

text: Motivation[edit]

text: Imagine a person walking over a landscape represented by the graph of y = f(x). Her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y. She is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L. If asked about the altitude of x = p, she would then answer L.

text: What, then, does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, her altitude is always ten meters or less from L.

text: The accuracy goal is then changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p, then her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.

text: The initial informal statement can now be explicated:

text: This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.

text: To say that

text: means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.

text: The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

text: Functions of a single variable[edit]

text: Suppose f : R → R is defined on the real line and p,L ∈ R. It is said the limit of f, as x approaches p, is L and written

text: if the following property holds:

text: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f(x) − L | < ε.

tag: ul

text: The value of the limit does not depend on the value of f(p), nor even that p be in the domain of f.

text: A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in R, and p a point of (a, b). Let f be a real-valued function defined on all of (a, b) except possibly at p itself. It is then said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ (a, b) implies | f(x) − L | < ε.

text: Here again the limit does not depend on f(p) being well-defined.

text:
The letters ε and δ can be understood as "error" and "distance", and in fact Cauchy used ε as an abbreviation for "error" in some of his work (Grabiner 1983), though in his definition of continuity he used an infinitesimal

α

{\displaystyle \alpha }

 rather than either ε or δ (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired by reducing the distance (δ) to the limit point. As discussed below this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.

text: Existence and one-sided limits[edit]

text: Alternatively x may approach p from above (right) or below (left), in which case the limits may be written as

text: or

text: respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, there is no limit at p (the limit at p does not exist). If either one-sided limit does not exist at p, the limit at p does not exist.

text: A formal definition is as follows. The limit of f(x) as x approaches p from above is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < x − p < δ. The limit of f(x) as x approaches p from below is L if, for every ε > 0, there exists a δ > 0 such that |f(x) − L| < ε whenever 0 < p − x < δ.

text: If the limit does not exist then the oscillation of f at p is non-zero.

text: More general subsets[edit]

text: Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows. Let f be a real-valued function defined on a subset S of the real line. Let p be a limit point of S—that is, p is the limit of some sequence of distinct elements of S. The limit of f, as x approaches p from values in S, is L if, for every ε > 0, there exists a δ > 0 such that 0 < |x − p| < δ and x ∈ S implies |f(x) − L| < ε.

text: This limit is often written

text:
The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form

(
−
∞
,
a
)

{\displaystyle (-\infty ,a)}

), and right-handed limits (e.g., by taking S to be an open interval of the form

(
a
,
∞
)

{\displaystyle (a,\infty )}

).

text: Deleted versus non-deleted limits[edit]

text: The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967), refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f:

text: A number L is the non-deleted limit of f as x approaches p if, for every ε > 0, there exists a δ > 0 such that | x − p | < δ and x ∈ Dm(f) implies | f(x) − L | < ε.

text: The definition is the same, except that the neighborhood | x − p | < δ now includes the point p, in contrast to the (deleted) neighborhood 0 < | x − p | < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits) (Hubbard (2015)).

text: Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) all by "limit" mean the deleted version.

text: Examples[edit]

text: Non-existence of one-sided limit(s)[edit]

text: The function

text:
has no limit at

x

0

=
1

{\displaystyle x_{0}=1}

 (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other x-coordinate.

text: The function

text: (the Dirichlet function) has no limit at any x-coordinate.

text: Non-equality of one-sided limits[edit]

text: The function

text: has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

text: Limits at only one point[edit]

text: The functions

text: and

text: both have a limit at x = 0 and it equals 0.

text: Limits at countably many points[edit]

text: The function

text:
has a limit at any x-coordinate of the form

π
2

+
2
n
π

{\displaystyle {\frac {\pi }{2}}+2n\pi }

, where n is any integer.

text: Functions on metric spaces[edit]

text: Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write

text: if the following property holds:

text: For every ε > 0, there exists a δ > 0 such that dB(f(x), L) < ε whenever 0 < dA(x, p) < δ.

text: Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.

text: An alternative definition using the concept of neighbourhood is as follows:

text: if, for every neighbourhood V of L in B, there exists a neighbourhood U of p in A such that f(U ∩ M − {p}) ⊆ V.

text: Functions on topological spaces[edit]

text: Suppose X,Y are topological spaces with Y a Hausdorff space. Let p be a limit point of Ω ⊆ X, and L ∈Y. For a function f : Ω → Y, it is said that the limit of f as x approaches p is L (i.e., f(x) → L as x → p) and written

text: if the following property holds:

text: For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.

text: This last part of the definition can also be phrased "there exists an open punctured neighbourhood U of p such that f(U∩Ω) ⊆ V ".

text: Note that the domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X − {p} (or all of X), then the limit of f as x → p exists and is equal to L if, for all subsets Ω of X with limit point p, the limit of the restriction of f to Ω exists and is equal to L. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on R by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.

text: Alternatively, the requirement that Y be a Hausdorff space can be relaxed to the assumption that Y be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.

text: A function is continuous in a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p.

text: Limits involving infinity[edit]

text: Limits at infinity[edit]

text: For f(x) a real function, the limit of f as x approaches infinity is L, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

, there exists c such that

|

f
(
x
)
−
L

|

<
ε

{\displaystyle |f(x)-L|<\varepsilon }

 whenever x > c. Or, symbolically:

text: Similarly, the limit of f as x approaches negative infinity is L, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

 there exists c such that

|

f
(
x
)
−
L

|

<
ε

{\displaystyle |f(x)-L|<\varepsilon }

 whenever x < c. Or, symbolically:

text: For example

text: Infinite limits[edit]

text: Limits can also have infinite values. When infinities are not considered legitimate values, which is standard (but see below), a formalist will insist upon various circumlocutions. For example, rather than say that a limit is infinity, the proper thing is to say that the function "diverges" or "grows without bound". In particular, the following informal example of how to pronounce the notation is arguably inappropriate in the classroom (or any other formal setting). In any case, for example the limit of f as x approaches a is infinity, denoted

text:
means that for all

ε
>
0

{\displaystyle \varepsilon >0}

 there exists

δ
>
0

{\displaystyle \delta >0}

 such that

f
(
x
)
>
ε

{\displaystyle f(x)>\varepsilon }

 whenever

|

x
−
a

|

<
δ

{\displaystyle |x-a|<\delta }

.

text: These ideas can be combined in a natural way to produce definitions for different combinations, such as

text: For example

text: Limits involving infinity are connected with the concept of asymptotes.

text: These notions of a limit attempt to provide a metric space interpretation to limits at infinity. However, note that these notions of a limit are consistent with the topological space definition of limit if

text:
a neighborhood of −∞ is defined to contain an interval [−∞, c) for some c ∈ R
a neighborhood of ∞ is defined to contain an interval (c, ∞] where c ∈ R
a neighborhood of a∈R is defined in the normal way metric space R

text: In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

text: Alternative notation[edit]

text:
Many authors[1] allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as R ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} where a neighborhood of ∞ is a set of the form {x: |x| > c}. The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line,

x

−
1

{\displaystyle x^{-1}}

 does not possess a central limit (which is normal):

text: In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

text:
In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of

lim

x
→

0

−

x

−
1

=
−
∞

{\displaystyle \lim _{x\to 0^{-}}{x^{-1}}=-\infty }

, namely, it is convenient for

lim

x
→
−
∞

x

−
1

=
−
0

{\displaystyle \lim _{x\to -\infty }{x^{-1}}=-0}

 to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

text: Limits at infinity for rational functions[edit]

text: There are three basic rules for evaluating limits at infinity for a rational function f(x) = p(x)/q(x): (where p and q are polynomials):

text:
If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
If the degree of p is less than the degree of q, the limit is 0.

text: If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

text: Functions of more than one variable[edit]

text: By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f : R2 → R,

text: if

text: where ||(x,y) − (p,q)|| represents the Euclidean distance. This can be extended to any number of variables.

text: Sequential limits[edit]

text: Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X and L ∈ Y.

text: If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general. If in addition X is metrizable, then L is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p.

text: Other characterizations[edit]

text: In terms of sequences[edit]

text: For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:

text:
if and only if for all sequences

x

n

{\displaystyle x_{n}}

 (with

x

n

{\displaystyle x_{n}}

 not equal to a for all n) converging to

a

{\displaystyle a}

 the sequence

f
(

x

n

)

{\displaystyle f(x_{n})}

 converges to

L

{\displaystyle L}

. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence

x

n

{\displaystyle x_{n}}

 to converge to

a

{\displaystyle a}

 requires the epsilon, delta method.

text:
Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f). Let a be the limit of a sequence of elements of Dm(f). Then the limit (in this sense) of f is L as x approaches p if for every sequence

x

n

{\displaystyle x_{n}}

 ∈ Dm(f) \ {a} (so that for all n,

x

n

{\displaystyle x_{n}}

 is not equal to a) that converges to a, the sequence

f
(

x

n

)

{\displaystyle f(x_{n})}

 converges to

L

{\displaystyle L}

. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f) of R as a metric space with the induced metric.

text: In non-standard calculus[edit]

text: In non-standard calculus the limit of a function is defined by:

text:
if and only if for all

x
∈

R

∗

{\displaystyle x\in \mathbb {R} ^{*}}

,

f

∗

(
x
)
−
L

{\displaystyle f^{*}(x)-L}

 is infinitesimal whenever

x
−
a

{\displaystyle x-a}

 is infinitesimal. Here

R

∗

{\displaystyle \mathbb {R} ^{*}}

 are the hyperreal numbers and

f

∗

{\displaystyle f^{*}}

 is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[2] On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.[3] Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[4]

text: In terms of nearness[edit]

text:
At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness". A point

x

{\displaystyle x}

 is defined to be near a set

A
⊆

R

{\displaystyle A\subseteq \mathbb {R} }

 if for every

r
>
0

{\displaystyle r>0}

 there is a point

a
∈
A

{\displaystyle a\in A}

 so that

|

x
−
a

|

<
r

{\displaystyle |x-a|<r}

. In this setting the

text:
if and only if for all

A
⊆

R

{\displaystyle A\subseteq \mathbb {R} }

,

L

{\displaystyle L}

 is near

f
(
A
)

{\displaystyle f(A)}

 whenever

a

{\displaystyle a}

 is near

A

{\displaystyle A}

. Here

f
(
A
)

{\displaystyle f(A)}

 is the set

{
f
(
x
)

|

x
∈
A
}

{\displaystyle \{f(x)|x\in A\}}

. This definition can also be extended to metric and topological spaces.

text: Relationship to continuity[edit]

text: The notion of the limit of a function is very closely related to the concept of continuity. A function ƒ is said to be continuous at c if it is both defined at c and its value at c equals the limit of f as x approaches c:

text: (We have here assumed that c is a limit point of the domain of f.)

text: Properties[edit]

text: If a function f is real-valued, then the limit of f at p is L if and only if both the right-handed limit and left-handed limit of f at p exist and are equal to L.

text: The function f is continuous at p if and only if the limit of f(x) as x approaches p exists and is equal to f(p). If f : M → N is a function between metric spaces M and N, then it is equivalent that f transforms every sequence in M which converges towards p into a sequence in N which converges towards f(p).

text: If N is a normed vector space, then the limit operation is linear in the following sense: if the limit of f(x) as x approaches p is L and the limit of g(x) as x approaches p is P, then the limit of f(x) + g(x) as x approaches p is L + P. If a is a scalar from the base field, then the limit of af(x) as x approaches p is aL.

text: If f is a real-valued (or complex-valued) function, then taking the limit is compatible with the algebraic operations, provided the limits on the right sides of the equations below exist (the last identity only holds if the denominator is non-zero). This fact is often called the algebraic limit theorem.

text: In each case above, when the limits on the right do not exist, or, in the last case, when the limits in both the numerator and the denominator are zero, nonetheless the limit on the left, called an indeterminate form, may still exist—this depends on the functions f and g. These rules are also valid for one-sided limits, for the case p = ±∞, and also for infinite limits using the rules

text:
q + ∞ = ∞ for q ≠ −∞
q × ∞ = ∞ if q > 0
q × ∞ = −∞ if q < 0
q / ∞ = 0 if q ≠ ± ∞

text: (see extended real number line).

text: Note that there is no general rule for the case q / 0; it all depends on the way 0 is approached. Indeterminate forms—for instance, 0/0, 0×∞, ∞−∞, and ∞/∞—are also not covered by these rules, but the corresponding limits can often be determined with L'Hôpital's rule or the Squeeze theorem.

text: Limits of compositions of functions[edit]

text: In general, from knowing that

text:
it does not follow that

lim

x
→
a

f
(
g
(
x
)
)
=
c

{\displaystyle \lim _{x\to a}f(g(x))=c}

. However, this "chain rule" does hold if one of the following additional conditions holds:

text:
f(b) = c (that is, f is continuous at b), or
g does not take the value b near a (that is, there exists a

δ
>
0

{\displaystyle \delta >0}

 such that if

0
<

|

x
−
a

|

<
δ

{\displaystyle 0<|x-a|<\delta }

 then

|

g
(
x
)
−
b

|

>
0

{\displaystyle |g(x)-b|>0}

).

text: As an example of this phenomenon, consider the following functions that violates both additional restrictions:

text: Since the value at f(0) is a removable discontinuity,

text: Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that

text: and so

text: Limits of special interest[edit]

text: Rational functions[edit]

text:
For

n

{\displaystyle n}

 a nonnegative integer and constants

a

1

,

a

2

,

a

3

,
…
,

a

n

{\displaystyle a_{1},a_{2},a_{3},\ldots ,a_{n}}

 and

b

1

,

b

2

,

b

3

,
…
,

b

n

{\displaystyle b_{1},b_{2},b_{3},\ldots ,b_{n}}

,

text:
lim

x
→
∞

a

1

x

n

+

a

2

x

n
−
1

+

a

3

x

n
−
2

+
.
.
.
+

a

n

b

1

x

n

+

b

2

x

n
−
1

+

b

3

x

n
−
2

+
.
.
.
+

b

n

=

a

1

b

1

{\displaystyle \lim _{x\to \infty }{\frac {a_{1}{x}^{n}+a_{2}{x}^{n-1}+a_{3}{x}^{n-2}+...+a_{n}}{b_{1}{x}^{n}+b_{2}{x}^{n-1}+b_{3}{x}^{n-2}+...+b_{n}}}={\frac {a_{1}}{b_{1}}}}

text:
This can be proven by dividing both the numerator and denominator by

x

n

{\displaystyle x^{n}}

. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

text: Trigonometric functions[edit]

text:
lim

x
→
0

sin

x

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}

lim

x
→
0

1
−
cos

x

x

=
0

{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}

lim

x
→
∞

x
sin

(

1
x

)

=
1

{\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1}

text: Exponential functions[edit]

text:
lim

x
→
0

(
1
+
x

)

1
x

=

lim

r
→
∞

(
1
+

1
r

)

r

=
e

{\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}=\lim _{r\to \infty }\left(1+{\frac {1}{r}}\right)^{r}=e}

lim

x
→
0

e

x

−
1

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}

lim

x
→
0

e

a
x

−
1

b
x

=

a
b

{\displaystyle \lim _{x\to 0}{\frac {e^{ax}-1}{bx}}={\frac {a}{b}}}

lim

x
→
0

c

a
x

−
1

b
x

=

a
b

ln

c

{\displaystyle \lim _{x\to 0}{\frac {c^{ax}-1}{bx}}={\frac {a}{b}}\ln c}

lim

x
→

0

+

x

x

=
1

{\displaystyle \lim _{x\to 0^{+}}x^{x}=1}

text: Logarithmic functions[edit]

text:
lim

x
→
0

ln

(
1
+
x
)

x

=
1

{\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}

lim

x
→
0

ln

(
1
+
a
x
)

b
x

=

a
b

{\displaystyle \lim _{x\to 0}{\frac {\ln(1+ax)}{bx}}={\frac {a}{b}}}

lim

x
→
0

log

c

(
1
+
a
x
)

b
x

=

a

b
ln

c

{\displaystyle \lim _{x\to 0}{\frac {\log _{c}(1+ax)}{bx}}={\frac {a}{b\ln c}}}

text: L'Hôpital's rule[edit]

text: This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:

text:
lim

x
→
c

f
(
x
)
=

lim

x
→
c

g
(
x
)
=
0
,

{\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0,}

 or

lim

x
→
c

f
(
x
)
=
±

lim

x
→
c

g
(
x
)
=
±
∞

{\displaystyle \lim _{x\to c}f(x)=\pm \lim _{x\to c}g(x)=\pm \infty }

, and

f

{\displaystyle f}

 and

g

{\displaystyle g}

 are differentiable over

I
∖
{
c
}

{\displaystyle I\setminus \{c\}}

, and

g
′

(
x
)
≠
0

{\displaystyle g'(x)\neq 0}

 for all

x
∈
I
∖
{
c
}

{\displaystyle x\in I\setminus \{c\}}

, and

lim

x
→
c

f
′

(
x
)

g
′

(
x
)

{\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}

 exists,

text: then:

text:
lim

x
→
c

f
(
x
)

g
(
x
)

=

lim

x
→
c

f
′

(
x
)

g
′

(
x
)

{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}

text: Normally, the first condition is the most important one.

text:
For example:

lim

x
→
0

sin

(
2
x
)

sin

(
3
x
)

=

lim

x
→
0

2
cos

(
2
x
)

3
cos

(
3
x
)

=

2
⋅
1

3
⋅
1

=

2
3

.

{\displaystyle \lim _{x\to 0}{\frac {\sin(2x)}{\sin(3x)}}=\lim _{x\to 0}{\frac {2\cos(2x)}{3\cos(3x)}}={\frac {2\cdot 1}{3\cdot 1}}={\frac {2}{3}}.}

text: Summations and integrals[edit]

text: Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

text:
A short way to write the limit

lim

n
→
∞

∑

i
=
s

n

f
(
i
)

{\displaystyle \lim _{n\to \infty }\sum _{i=s}^{n}f(i)}

 is

∑

i
=
s

∞

f
(
i
)

{\displaystyle \sum _{i=s}^{\infty }f(i)}

. An important example of limits of sums such as these are series.

text:
A short way to write the limit

lim

x
→
∞

∫

a

x

f
(
t
)

d
t

{\displaystyle \lim _{x\to \infty }\int _{a}^{x}f(t)\;dt}

 is

∫

a

∞

f
(
t
)

d
t

{\displaystyle \int _{a}^{\infty }f(t)\;dt}

.

text:
A short way to write the limit

lim

x
→
−
∞

∫

x

b

f
(
t
)

d
t

{\displaystyle \lim _{x\to -\infty }\int _{x}^{b}f(t)\;dt}

 is

∫

−
∞

b

f
(
t
)

d
t

{\displaystyle \int _{-\infty }^{b}f(t)\;dt}

.


document_title: History of rugby union matches between England and Ireland

question_text: how many times have ireland beaten england in the six nations

text: Breakdown[edit]

text: In the Home Nations Championship (1874–1909 and 1932–1939), England played Ireland on 41 occasions, England winning 27, Ireland 13 with one match having been drawn.

text: In the Five Nations (1910–1931 and 1947–1999), England played Ireland on 70 occasions, England winning 38, Ireland 25 with seven matches having been drawn. In these matches, England have scored 851 points, and Ireland 575.

text: In the Six Nations (2000–present), England have played Ireland on 19 occasions, England winning 8, Ireland 11 and no matches have been drawn. In these matches, England have scored 384 points, and Ireland 310.

text: In other test matches, England have played Ireland on three occasions (1988, 2011 and 2015), with England winning all three. In these matches, England have scored 62 points, and Ireland 32.

text: Summary[edit]

text: Overall[edit]

text:
Details
Played
Won by
 England
Won by
 Ireland
Drawn
England points
Ireland points

In Ireland
67
32
31
4
635
599

In England
66
44
18
4
915
517

Neutral venue
0
0
0
0
0
0

Overall
133
76
49
8
1550
1116

tag: table

text: Records[edit]

text: Note: Date shown in brackets indicates when the record was or last set.

text:
Record
England
Ireland

Longest winning streak
8 (10 Feb 1912–14 Feb 1925)
5 (12 Feb 1972–5 Feb 1977)

Largest points for

Home
50 (5 February 2000)
43 (24 February 2007)

Away
46 (15 February 1997)
28 (18 March 2006)

Largest winning margin

Home
34 (16 February 2002)
30 (24 February 2007)

Away
40 (15 February 1997)
13 (8 February 1964)

text: Results[edit]

text:
No.
Date
Venue
Score
Winner
Competition
Match report

1
15 Feb 1875
The Oval, London
2G – 0
 England
1874–75 Home Nations rugby union matches

2
13 Dec 1875
Rathmines, Dublin
0 – 1G
 England
1875–76 Home Nations rugby union matches

3
5 Feb 1877
The Oval , London
2G – 0
 England
1876–77 Home Nations rugby union matches

4
11 Mar 1878
Lansdowne Road, Dublin
0 – 2G
 England
1877–78 Home Nations rugby union matches

5
24 Mar 1879
The Oval, London
3G – 0
 England
1878–79 Home Nations rugby union matches

6
30 Jan 1880
Lansdowne Road, Dublin
0 – 1G
 England
1879–80 Home Nations rugby union matches

7
10 Feb 1881
Whalley Range, Manchester
2G – 0
 England
1880–81 Home Nations rugby union matches

8
6 Feb 1882
Lansdowne Road, Dublin
0 – 0
draw
1881–82 Home Nations rugby union matches

9
10 Feb 1883
Whalley Range, Manchester
1G – 0
 England
1883 Home Nations Championship

10
4 Feb 1884
Lansdowne Road, Dublin
0 – 1G
 England
1884 Home Nations Championship

11
7 Feb 1885
Whalley Range, Manchester
(2) 0G – 0G (1)
 England
1885 Home Nations Championship

12
6 Feb 1886
Lansdowne Road, Dublin
0 – 0G (1)
 England
1886 Home Nations Championship

13
5 Feb 1887
Lansdowne Road, Dublin
2G – 0
 Ireland
1887 Home Nations Championship

14
15 Mar 1890
Rectory Field, London
3 – 0
 England
1890 Home Nations Championship

15
7 Feb 1891
Lansdowne Road, Dublin
0 – 9
 England
1891 Home Nations Championship

16
6 Feb 1892
Whalley Range, Manchester
7 – 0
 England
1892 Home Nations Championship

17
4 Feb 1893
Lansdowne Road, Dublin
0 – 4
 England
1893 Home Nations Championship

18
3 Feb 1894
Rectory Field, London
5 – 7
 Ireland
1894 Home Nations Championship

19
2 Feb 1895
Lansdowne Road, Dublin
3 – 6
 England
1895 Home Nations Championship

20
1 Feb 1896
Meanwood Road, Leeds, Leeds
4 – 10
 Ireland
1896 Home Nations Championship

21
6 Feb 1897
Lansdowne Road, Dublin
13 – 9
 Ireland
1897 Home Nations Championship

22
5 Feb 1898
Athletic Ground, Richmond
6 – 9
 Ireland
1898 Home Nations Championship

23
4 Feb 1899
Lansdowne Road, Dublin
6 – 0
 Ireland
1899 Home Nations Championship

24
3 Feb 1900
Athletic Ground, Richmond
15 – 4
 England
1900 Home Nations Championship

25
9 Feb 1901
Lansdowne Road, Dublin
10 – 6
 Ireland
1901 Home Nations Championship

26
8 Feb 1902
Welford Road, Leicester
6 – 3
 England
1902 Home Nations Championship

27
14 Feb 1903
Lansdowne Road, Dublin
6 – 0
 Ireland
1903 Home Nations Championship

28
13 Feb 1904
Rectory Field, London
19 – 0
 England
1904 Home Nations Championship

29
11 Feb 1905
Mardyke, Cork
17 – 3
 Ireland
1905 Home Nations Championship

30
10 Feb 1906
Welford Road, Leicester
6 – 16
 Ireland
1906 Home Nations Championship

31
9 Feb 1907
Lansdowne Road, Dublin
17 – 9
 Ireland
1907 Home Nations Championship

32
8 Feb 1908
Athletic Ground, Richmond
13 – 3
 England
1908 Home Nations Championship

33
13 Feb 1909
Lansdowne Road, Dublin
5 – 11
 England
1909 Home Nations Championship

34
11 Feb 1910
Twickenham, London
0 – 0
draw
1910 Five Nations Championship

35
11 Feb 1911
Lansdowne Road, Dublin
3 – 0
 Ireland
1911 Five Nations Championship

36
10 Feb 1912
Twickenham, London
15 – 0
 England
1912 Five Nations Championship

37
9 Feb 1913
Lansdowne Road, Dublin
4 – 15
 England
1913 Five Nations Championship

38
14 Feb 1914
Twickenham, London
17 – 12
 England
1914 Five Nations Championship

39
14 Feb 1920
Lansdowne Road, Dublin
11 – 14
 England
1920 Five Nations Championship

40
12 Feb 1921
Twickenham, London
15 – 0
 England
1921 Five Nations Championship

41
11 Feb 1922
Lansdowne Road, Dublin
3 – 12
 England
1922 Five Nations Championship

42
10 Feb 1923
Welford Road, Leicester
23 – 5
 England
1923 Five Nations Championship

43
9 Feb 1924
Ravenhill, Belfast
3 – 14
 England
1924 Five Nations Championship

44
14 Feb 1925
Twickenham, London
6 – 6
draw
1925 Five Nations Championship

45
13 Feb 1926
Lansdowne Road, Dublin
19 – 15
 Ireland
1926 Five Nations Championship

46
12 Feb 1927
Twickenham, London
8 – 6
 England
1927 Five Nations Championship

47
11 Feb 1928
Lansdowne Road, Dublin
6 – 7
 England
1928 Five Nations Championship

48
9 Feb 1929
Twickenham, London
5 – 6
 Ireland
1929 Five Nations Championship

49
8 Feb 1930
Lansdowne Road, Dublin
4 – 3
 Ireland
1930 Five Nations Championship

50
14 Feb 1931
Twickenham, London
5 – 6
 Ireland
1931 Five Nations Championship

51
13 Feb 1932
Lansdowne Road, Dublin
8 – 11
 England
1932 Home Nations Championship

52
11 Feb 1933
Twickenham, London
17 – 6
 England
1933 Home Nations Championship

53
10 Feb 1934
Lansdowne Road, Dublin
3 – 13
 England
1934 Home Nations Championship

54
9 Feb 1935
Twickenham, London
14 – 3
 England
1935 Home Nations Championship

55
8 Feb 1936
Lansdowne Road, Dublin
6 – 3
 Ireland
1936 Home Nations Championship

56
13 Feb 1937
Twickenham, London
9 – 8
 England
1937 Home Nations Championship

57
12 Feb 1938
Lansdowne Road, Dublin
14 – 36
 England
1938 Home Nations Championship

58
11 Feb 1939
Twickenham, London
0 – 5
 Ireland
1939 Home Nations Championship

59
8 Feb 1947
Lansdowne Road, Dublin
22 – 0
 Ireland
1947 Five Nations Championship

60
14 Feb 1948
Twickenham, London
10 – 11
 Ireland
1948 Five Nations Championship
Clip

61
12 Feb 1949
Lansdowne Road, Dublin
14 – 5
 Ireland
1949 Five Nations Championship

62
11 Feb 1950
Twickenham, London
3 – 0
 England
1950 Five Nations Championship

63
10 Feb 1951
Lansdowne Road, Dublin
3 – 0
 Ireland
1951 Five Nations Championship

64
29 Mar 1952
Twickenham, London
3 – 0
 England
1952 Five Nations Championship

65
11 Feb 1953
Lansdowne Road, Dublin
9 – 9
draw
1953 Five Nations Championship

66
13 Feb 1954
Twickenham, London
14 – 3
 England
1954 Five Nations Championship

67
11 Feb 1955
Lansdowne Road, Dublin
6 – 6
draw
1955 Five Nations Championship

68
10 Feb 1956
Twickenham, London
20 – 0
 England
1956 Five Nations Championship

69
9 Feb 1957
Lansdowne Road, Dublin
0 – 6
 England
1957 Five Nations Championship

70
08 Feb 1958
Twickenham, London
6 – 0
 England
1958 Five Nations Championship

71
14 Feb 1959
Lansdowne Road, Dublin
0 – 3
 England
1959 Five Nations Championship

72
13 Feb 1960
Twickenham, London
8 – 5
 England
1960 Five Nations Championship

73
11 Feb 1961
Lansdowne Road, Dublin
11 – 8
 Ireland
1961 Five Nations Championship

74
10 Feb 1962
Twickenham, London
16 – 0
 England
1962 Five Nations Championship

75
11 Feb 1963
Lansdowne Road, Dublin
0 – 0
draw
1963 Five Nations Championship

76
08 Feb 1964
Twickenham, London
5 – 18
 Ireland
1964 Five Nations Championship

77
13 Feb 1965
Lansdowne Road, Dublin
5 – 0
 Ireland
1965 Five Nations Championship

78
12 Feb 1966
Twickenham, London
6 – 6
draw
1966 Five Nations Championship

79
11 Feb 1967
Lansdowne Road, Dublin
3 – 8
 England
1967 Five Nations Championship

80
10 Feb 1968
Twickenham, London
9 – 9
draw
1968 Five Nations Championship

81
8 Feb 1969
Lansdowne Road, Dublin
17 – 15
 Ireland
1969 Five Nations Championship

82
14 Feb 1970
Twickenham, London
9 – 3
 England
1970 Five Nations Championship

83
13 Feb 1971
Lansdowne Road, Dublin
6 – 9
 England
1971 Five Nations Championship

84
12 Feb 1972
Twickenham, London
12 – 16
 Ireland
1972 Five Nations Championship

85
10 Feb 1973
Lansdowne Road, Dublin
18 – 9
 Ireland
1973 Five Nations Championship

86
06 Feb 1974
Twickenham, London
21 – 26
 Ireland
1974 Five Nations Championship

87
18 Jan 1975
Lansdowne Road, Dublin
12 – 9
 Ireland
1975 Five Nations Championship

88
06 Mar 1976
Twickenham, London
12 – 13
 Ireland
1976 Five Nations Championship

89
05 Feb 1977
Lansdowne Road, Dublin
0 – 4
 England
1977 Five Nations Championship

90
18 Mar 1978
Twickenham, London
15 – 9
 England
1978 Five Nations Championship

91
17 Feb 1979
Lansdowne Road, Dublin
12 – 7
 Ireland
1979 Five Nations Championship

92
19 Jan 1980
Twickenham, London
24 – 9
 England
1980 Five Nations Championship

93
7 Mar 1981
Lansdowne Road, Dublin
6 – 10
 England
1981 Five Nations Championship

94
06 Feb 1982
Twickenham, London
15 – 16
 Ireland
1982 Five Nations Championship

95
19 Mar 1983
Lansdowne Road, Dublin
25 – 15
 Ireland
1983 Five Nations Championship

96
18 Feb 1984
Twickenham, London
12 – 9
 England
1984 Five Nations Championship

97
30 Mar 1985
Lansdowne Road, Dublin
13 – 10
 Ireland
1985 Five Nations Championship
IT

98
01 Mar 1986
Twickenham, London
25 – 20
 England
1986 Five Nations Championship

99
7 February 1987
Lansdowne Road, Dublin
17 – 0
 Ireland
1987 Five Nations Championship

100
19 March 1988
Twickenham, London
35 – 3
 England
1988 Five Nations Championship

101
23 April 1988
Lansdowne Road, Dublin
10 – 21
 England
Inaugural Millennium Trophy

102
18 February 1989
Lansdowne Road, Dublin
3 – 16
 England
1989 Five Nations Championship

103
20 January 1990
Twickenham, London
23 – 0
 England
1990 Five Nations Championship

104
2 March 1991
Lansdowne Road, Dublin
7 – 16
 England
1991 Five Nations Championship

105
1 February 1992
Twickenham, London
38 – 9
 England
1992 Five Nations Championship

106
20 March 1993
Lansdowne Road, Dublin
17 – 3
 Ireland
1993 Five Nations Championship
IT

107
19 February 1994
Twickenham, London
12 – 13
 Ireland
1994 Five Nations Championship

108
21 January 1995
Lansdowne Road, Dublin
8 – 20
 England
1995 Five Nations Championship

109
16 March 1996
Twickenham, London
28 – 15
 England
1996 Five Nations Championship

110
15 February 1997
Lansdowne Road, Dublin
6 – 46
 England
1997 Five Nations Championship

111
4 April 1998
Twickenham, London
35 – 17
 England
1998 Five Nations Championship
NYT

112
6 March 1999
Lansdowne Road, Dublin
15 – 27
 England
1999 Five Nations Championship
BBC

113
5 February 2000
Twickenham, London
50 – 18
 England
2000 Six Nations Championship
BBC

114
20 October 2001
Lansdowne Road, Dublin
20 – 14
 Ireland
2001 Six Nations Championship
BBC

115
16 February 2002
Twickenham, London
45 – 11
 England
2002 Six Nations Championship
BBC, TG

116
30 March 2003
Lansdowne Road, Dublin
6 – 42
 England
2003 Six Nations Championship
BBC, TG

117
6 March 2004
Twickenham, London
13 – 19
 Ireland
2004 Six Nations Championship
BBC

118
27 February 2005
Lansdowne Road, Dublin
19 – 13
 Ireland
2005 Six Nations Championship
BBC

119
18 March 2006
Twickenham, London
24 – 28
 Ireland
2006 Six Nations Championship
BBC

120
24 February 2007
Croke Park, Dublin
43 – 13
 Ireland
2007 Six Nations Championship
BBC

121
15 March 2008
Twickenham, London
33 – 10
 England
2008 Six Nations Championship
BBC

122
28 February 2009
Croke Park, Dublin
14 – 13
 Ireland
2009 Six Nations Championship
BBC

123
27 February 2010
Twickenham, London
16 – 20
 Ireland
2010 Six Nations Championship
BBC

124
19 March 2011
Aviva Stadium, Dublin
24 – 8
 Ireland
2011 Six Nations Championship
BBC

125
27 August 2011
Aviva Stadium, Dublin
9 – 20
 England
2011 Rugby World Cup warm up
BBC

126
17 March 2012
Twickenham, London
30 – 9
 England
2012 Six Nations Championship
BBC

127
10 February 2013
Aviva Stadium, Dublin
6 – 12
 England
2013 Six Nations Championship
BBC

128
22 February 2014
Twickenham, London
13 – 10
 England
2014 Six Nations Championship
BBC

129
1 March 2015
Aviva Stadium, Dublin
19 – 9
 Ireland
2015 Six Nations Championship
BBC

130
5 September 2015
Twickenham, London
21 – 13
 England
2015 Rugby World Cup warm up
BBC

131
27 February 2016
Twickenham, London
21 – 10
 England
2016 Six Nations Championship
BBC

132
18 March 2017
Aviva Stadium, Dublin
13 – 9
 Ireland
2017 Six Nations Championship
BBC

133
17 March 2018
Twickenham, London
24 – 15
 Ireland
2018 Six Nations Championship
BBC