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Criminal Minds (season 13)
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Criminal Minds (season 13)
Promotional poster
Starring
Joe Mantegna
Matthew Gray Gubler
A.J. Cook
Kirsten Vangsness
Aisha Tyler
Daniel Henney
Adam Rodriguez
Paget Brewster
Country of origin
United States
No. of episodes
14
Release
Original network
CBS
Original release
September 27, 2017 (2017-09-27) – present
Season chronology
← Previous
Season 12
List of Criminal Minds episodes
The thirteenth season of Criminal Minds was ordered on April 7, 2017, by CBS with an order of 22 episodes.[1][2] The season premiered on September 27, 2017 in a new time slot at 10:00PM on Wednesday when it had previously been at 9:00PM on Wednesday since its inception. The season will conclude on April 25, 2018.[3]
Contents
[hide]
1 Cast
1.1 Main cast
1.2 Special guest star
1.3 Recurring cast
1.4 Guest cast
2 Production
2.1 Development
2.2 Casting
3 Episodes
4 Ratings
5 References
6 External links
Cast[edit]
The entire main cast from the previous season returned for the show, except Damon Gupton (Stephen Walker), who was fired from the show.[4][5] His character was killed off in the season premiere off-screen.
Following the cancellation of Criminal Minds: Beyond Borders, it was announced that Daniel Henney (Matt Simmons) would join the cast this season as a series regular.[6]
Main cast[edit]
Joe Mantegna as David Rossi
Matthew Gray Gubler as Dr. Spencer Reid
A. J. Cook as Jennifer "JJ" Jareau
Kirsten Vangsness as Penelope Garcia
Aisha Tyler as Dr. Tara Lewis
Daniel Henney as Matt Simmons
Adam Rodriguez as Luke Alvez
Paget Brewster as Emily Prentiss
Special guest star[edit]
Shemar Moore as Derek Morgan
Recurring cast[edit]
Josh Stewart as William LaMontagne Jr.
Kelly Frye as Kristy Simmons
Declan Whaley as David Simmons
Kim Rhodes as Assistant Director Linda Barnes
Guest cast[edit]
Bodhi Elfman as Peter Lewis / Mr. Scratch
Tracie Thoms as Monica Walker
Jamie Kennedy as Floyd Feylinn Ferell
Zelda Williams as Melissa Miller
Lou Diamond Phillips as Sheriff Clifford
Production[edit]
Development[edit]
Criminal Minds was renewed for a thirteenth season with an episode order of 22 episodes on April 7, 2017.[1][2] The entire main cast from the previous season returned for the season, except Damon Gupton (Stephen Walker), who was fired from the show.[4][5]
Matthew Gray Gubler will be directing the seventeenth episode of the season and it is said to be "the spookiest episode of Season 13" and will involve clowns.[7] On August 10, 2017, it was revealed that Aisha Tyler will make her television directing debut and direct the sixth episode of the season.[8] On August 12, 2017 it was revealed that Erica Messer and Kirsten Vangsness will be co-writing the eleventh episode of the season, which will be the fourth episode they have co-written together.[9] On October 31, 2017, it was announced that Adam Rodriguez will make his Criminal Minds directing debut and direct the sixteenth episode of the season.[10]
Casting[edit]
Shemar Moore reprised his role as Derek Morgan in the fifth episode.
On June 11, 2017, it was announced that Damon Gupton had been let go from the show after one season.[4] CBS said his departure was "part of a creative change on the show".[4][5] His character Stephen Walker was killed off-screen in the season premiere.
On June 20, 2017, CBS announced that Daniel Henney, who was a series regular on Criminal Minds: Beyond Borders as Matt Simmons, would join the main show as a series regular for the thirteenth season.[11]
On October 12, 2017, it was announced that Shemar Moore would reprise his role as Derek Morgan in the fifth episode of the season ("Lucky Strikes") as his character returns to help Penelope Garcia get through a tough time.[12]
Episodes[edit]
See also: List of Criminal Minds episodes
No.
overall
No. in
season
Title
Directed by
Written by
Original air date
Prod.
code
U.S. viewers
(millions)
278
1 "Wheels Up" Glenn Kershaw Breen Frazier September 27, 2017 (2017-09-27) 1301 7.00[13]
When escaped serial killer Peter Lewis (Bodhi Elfman) launches a devastating final assault on the BAU and takes one of their own as a hostage, Garcia brings in former International Response Team member Matt Simmons (Daniel Henney) to help apprehend Lewis before more lives are lost.
279
2 "To a Better Place" Diana C. Valentine Bruce Zimmerman October 4, 2017 (2017-10-04) 1302 6.17[14]
When suitcases containing human remains are found throughout Naples, Florida, the BAU returns from their bureau-dictated vacation to find a connection between the victims before the killer takes another life. Meanwhile, Reid learns about a condition to his reinstatement and the team officially welcomes Simmons into the fold.
280
3 "Blue Angel" Sharat Raju Christopher Barbour October 11, 2017 (2017-10-11) 1304 5.87[15]
When two businessmen are emasculated in Detroit, Michigan, the BAU investigates a connection between the victims only to discover a family secret that changes the course of the investigation. Meanwhile, Simmons finds himself bonding with an escort involved in the case.
281
4 "Killer App" Alec Smight Stephanie SenGupta October 18, 2017 (2017-10-18) 1303 5.94[16]
When a workplace shooting in Silicon Valley claims the lives of three employees, the BAU works with the Department of Homeland Security to determine if the shooting was personal and why the victims were targeted. Meanwhile, Prentiss confronts Alvez about the events surrounding his final confrontation with Peter Lewis.
282
5 "Lucky Strikes" Tawnia McKiernan Jim Clemente October 25, 2017 (2017-10-25) 1306 5.91[17]
When a Florida prostitute is found dead with both legs and several fingers missing, the BAU sets out to determine if cannibalistic serial killer Floyd Feylinn Ferell (Jamie Kennedy) has embarked on another spree or if a copycat killer is responsible. Meanwhile, former SSA Derek Morgan (Shemar Moore) helps Garcia when she becomes overwhelmed by memories of a ten-year-old trauma.
283
6 "The Bunker" Aisha Tyler Karen Maser November 8, 2017 (2017-11-08) 1305 5.50[18]
When a missing persons case is connected to a series of abductions throughout Virginia, the BAU sets out to find a connection between the victims only to unearth an unusual motive. Meanwhile, JJ connects with a woman who believes her sister's disappearance may be connected to the investigation and Garcia searches for a new apartment.
284
7 "Dust and Bones" Marcus Stokes Erica Meredith November 15, 2017 (2017-11-15) 1307 5.64[19]
When two women are abducted and permanently disfigured in Austin, Texas, the BAU determines that the attacker has a dangerous obsession with a vicious creature. Meanwhile, Alvez struggles to find a way to help his former Fugitive Task Force partner come to terms with his paralysis.
285
8 "Neon Terror" Bethany Rooney Erik Stiller November 22, 2017 (2017-11-22) 1308 6.31[20]
When the latest in a series of robbery-homicides occurs in Miami, Florida, the BAU investigates only to suspect that they are dealing with an unsub who seeks attention by releasing footage of his crimes to the media.
286
9 "False Flag" Joe Mantegna Breen Frazier December 6, 2017 (2017-12-06) 1309 5.34[21]
When two members of a Roswell, New Mexico conspiracy theory group die in quick succession, the BAU suspects that the crimes were committed by a serial killer only to find themselves reexamining the evidence following a catastrophic encounter. Meanwhile, Lewis struggles to keep her emotions in check as she interrogates a young podcaster (Zelda Williams) with ties to the group.
287
10 "Submerged" Rob Bailey Bruce Zimmerman January 3, 2018 (2018-01-03) 1310 5.41[22]
When the latest in a series of perplexing robbery-homicides occurs in Ramona, California, the BAU works with a frustrated sheriff (Lou Diamond Phillips) to track down a serial killer with mysterious motives and an even more mysterious past.
288
11 "Full-Tilt Boogie" Simon Mirren Erica Messer & Kirsten Vangsness January 10, 2018 (2018-01-10) 1311 5.70[23]
When the wife of a Virginia police chief survives a brutal home invasion-homicide, the BAU coordinates with her husband and the local authorities to catch her attempted killer only to uncover a multitude of secrets being kept by the town's inhabitants.
289
12 "Bad Moon on the Rise" Christoph Schrewe Karen Maser January 17, 2018 (2018-01-17) 1312 5.63[24]
When a New York City jogger's gruesome death is connected to a series of murder-mutilations, the BAU searches Central Park for a killer suffering from a syndrome that makes him believe he is a vicious creature. Meanwhile, Alvez deals with the fallout of a blind date gone wrong.
290
13 "Cure" Glenn Kershaw Christopher Barbour January 24, 2018 (2018-01-24) 1313 5.30[25]
When a cryptic message is found inside the mouth of a Washington, D.C. financier, the BAU sets out to catch a serial killer determined to eradicate institutions of power. Meanwhile, JJ grows increasingly concerned after receiving an email asking her to sit down with Assistant Director Linda Barnes (Kim Rhodes).
291
14 "Miasma" Leon Ichaso Erica Meredith January 31, 2018 (2018-01-31) 1314 5.42[26]
When a mass grave is uncovered in a New Orleans cemetery, the BAU searches for a killer who sees himself as a modern-day plague doctor. Meanwhile, Prentiss finds herself forced to endure a hostile interrogation after Barnes suspends her from active duty and summons her for an internal audit.
292
15 "Annihilator"[27] Rob Bailey Erik Stiller March 7, 2018 (2018-03-07) 1315 TBD
293
16 "Last Gasp"[28] Adam Rodriguez Stephanie SenGupta March 14, 2018 (2018-03-14) 1316 TBD
294
17 "The Capilano's"[29] Matthew Gray Gubler Erica Messer March 21, 2018 (2018-03-21) 1317 TBD
295
18 "TBA" Joe Mantegna[30] TBA March 28, 2018 (2018-03-28) 1318 TBD
296
19 "TBA" Lily Mariye[31] Christopher Barbour April 4, 2018 (2018-04-04) 1319 TBD
297
20 "All You Can Eat"[32] Diana Valentine Karen Maser April 11, 2018 (2018-04-11) 1320 TBD
298
21 "Mixed Signals"[33] Alec Smight Erica Meredith & Erik Stiller April 18, 2018 (2018-04-18) 1321 TBD
Ratings[edit]
No.
Title
Air date
Rating/share
(18–49)
Viewers
(millions)
DVR
(18–49)
DVR viewers
(millions)
Total
(18–49)
Total viewers
(millions)
1 "Wheels Up" September 27, 2017 1.3/5 7.00[13] 1.1 4.15 2.4 11.15[34]
2 "To a Better Place" October 4, 2017 1.1/4 6.17[14] 1.1 3.98 2.2 10.15[35]
3 "Blue Angel" October 11, 2017 1.1/4 5.87[15] 1.0 3.82 2.1 9.68[36]
4 "Killer App" October 18, 2017 1.0/4 5.94[16] 1.1 3.87 2.1 9.81[37]
5 "Lucky Strikes" October 25, 2017 1.0/4 5.91[17] 1.0 3.81 2.0 9.72[38]
6 "The Bunker" November 8, 2017 0.9/3 5.50[18] 1.0 3.53 1.9 9.03[39]
7 "Dust and Bones" November 15, 2017 1.0/4 5.64[19] 1.0 3.76 2.0 9.40[40]
8 "Neon Terror" November 22, 2017 1.0/4 6.31[20] 1.0 3.70 2.0 10.00[41]
9 "False Flag" December 6, 2017 0.9/4 5.34[21] 1.1 4.04 2.0 9.38[42]
10 "Submerged" January 3, 2018 0.9/3 5.41[22] 1.1 3.93 2.0 9.37[43]
11 "Full-Tilt Boogie" January 10, 2018 1.0/4 5.70[23] 1.1 4.13 2.1 9.83[44]
12 "Bad Moon on the Rise" January 17, 2018 1.0/4 5.63[24] 1.1 4.02 2.1 9.65[45]
13 "Cure" January 24, 2018 0.9/3 5.30[25] 1.1 4.12 2.0 9.42[46]
14 "Miasma" January 31, 2018 1.0/4 5.42[26] 1.0 3.90 2.0 9.32[47]
15 "Annihilator" March 7, 2018 TBD TBD TBD TBD TBD TBD
16 "Last Gasp" March 14, 2018 TBD TBD TBD TBD TBD TBD
17 "The Capilano's" March 21, 2018 TBD TBD TBD TBD TBD TBD
References[edit]
^ Jump up to: a b Andreeva, Nellie (April 7, 2017). "'Criminal Minds' Renewed For Season 13 By CBS". Deadline Hollywood. Retrieved April 7, 2017.
^ Jump up to: a b Angelica, Patricia (May 4, 2017). "The Replicator, season 13, writers' strike, USC: an exclusive interview with Harry Bring, the LLPOS of Criminal Minds". Criminal Minds Brasil. Retrieved May 17, 2017.
Jump up ^ "CRIMINAL MINDS Listings". The Futon Critic. Retrieved January 14, 2018.
^ Jump up to: a b c d Andreeva, Nellie (June 11, 2017). "'Criminal Minds': Damon Gupton Leaving CBS Series After One Season". TVLine. Retrieved June 11, 2017.
^ Jump up to: a b c Gupton, Damon [@DamonGupton] (June 10, 2017). "wanted to let you know I appreciate this as I just lost my job. You throwing out the 1st pitch is pure class and a monumental image" (Tweet). Retrieved June 10, 2017 – via Twitter.
Jump up ^ Andreeva, Nellie (June 20, 2017). "'Criminal Minds': Daniel Henney Of 'Beyond Borders' Joins Mothership Series". TVLine. Retrieved June 20, 2017.
Jump up ^ Ausiello, Michael (August 2, 2017). "Ask Ausiello: Spoilers on X-Files, Riverdale, Supergirl, Supernatural, Chuck, NCIS, Elementary and More". TV Guide. Retrieved August 2, 2017.
Jump up ^ Bring, Harry [@LLPOS] (August 10, 2017). "Today starts 1305 prep. Director: Aisha Tyler Writer: Karen Maser "The Bunker"" (Tweet). Retrieved November 2, 2017 – via Twitter.
Jump up ^ Maser, Karen [@KarenMaser] (August 12, 2017). "Then you will be happy to watch Ep 1311 to be written by the fabulous @ecmess & @Vangsness" (Tweet). Retrieved August 12, 2017 – via Twitter.
Jump up ^ Maser, Karen [@KarenMaser] (October 31, 2017). "#1316. It will be written by Stephanie Sengupta" (Tweet). Retrieved November 2, 2017 – via Twitter.
Jump up ^ "Criminal Minds Adds Beyond Borders ' Daniel Henney as Series Regular". TVLine.com. Retrieved June 20, 2017.
Jump up ^ Webb Mitovich, Matt (October 12, 2017). "Criminal Minds: Shemar Moore Set for Encore, to Help Garcia With [Spoiler]". TVLine. Retrieved October 19, 2017.
^ Jump up to: a b Porter, Rick (September 28, 2017). "'Survivor' premiere adjusts up: Wednesday final ratings". TV by the Numbers. Retrieved September 28, 2017.
^ Jump up to: a b Porter, Rick (October 5, 2017). "'SEAL Team,' 'Law & Order: SVU' and everything else unchanged: Wednesday final ratings". TV by the Numbers. Retrieved October 5, 2017.
^ Jump up to: a b Porter, Rick (October 12, 2017). "'Empire,' 'Survivor,' 'Goldbergs,' 'Blacklist' adjust up, 'Dynasty' adjusts down: Wednesday final ratings". TV by the Numbers. Retrieved October 12, 2017.
^ Jump up to: a b Porter, Rick (October 19, 2017). "'Survivor,' 'Goldbergs,' 'SEAL Team' adjust up, 'Criminal Minds' adjusts down: Wednesday final ratings". TV by the Numbers. Retrieved October 19, 2017.
^ Jump up to: a b Porter, Rick (October 26, 2017). "'The Blacklist' and 'Designated Survivor' adjust down, final World Series numbers: Wednesday final ratings". TV by the Numbers. Retrieved October 26, 2017.
^ Jump up to: a b Porter, Rick (November 9, 2017). "'Empire,' 'SVU,' CMAs and others unchanged: Wednesday final ratings". TV by the Numbers. Retrieved November 9, 2017.
^ Jump up to: a b Porter, Rick (November 16, 2017). "'The Goldbergs' adjusts up: Wednesday final ratings". TV by the Numbers. Retrieved November 16, 2017.
^ Jump up to: a b Porter, Rick (November 27, 2017). "'The Wall' and 'SNL Thanksgiving' adjust up: Wednesday final ratings". TV by the Numbers. Retrieved November 27, 2017.
^ Jump up to: a b Porter, Rick (December 7, 2017). "'Survivor,' 'SVU' and everything else unchanged: Wednesday final ratings". TV by the Numbers. Retrieved December 7, 2017.
^ Jump up to: a b Porter, Rick (January 5, 2018). "'Chicago PD' adjusts up, 'The X-Files' adjusts down: Wednesday final ratings". TV by the Numbers. Retrieved January 5, 2018.
^ Jump up to: a b Porter, Rick (January 11, 2018). "'Speechless' adjusts up: Wednesday final ratings". TV by the Numbers. Retrieved January 11, 2018.
^ Jump up to: a b Porter, Rick (January 19, 2018). "'Riverdale,' 'X-Files' and everything else unchanged: Wednesday final ratings". TV by the Numbers. Retrieved January 19, 2018.
^ Jump up to: a b Porter, Rick (January 25, 2018). "'Match Game' adjusts up: Wednesday final ratings". TV by the Numbers. Retrieved January 25, 2018.
^ Jump up to: a b Porter, Rick (February 1, 2018). "'The Amazing Race' adjusts up, 'The Blacklist' adjusts down: Wednesday final ratings". TV by the Numbers. Retrieved February 1, 2018.
Jump up ^ Bring, Harry [@LLPOS] (December 14, 2017). "1315 "Annihilator" Writer Erik Stiller Director Rob Bailey. Shoots 3 days before our holiday hiatus and then 5 days after we come back to work Jan 4th" (Tweet). Retrieved December 14, 2017 – via Twitter.
Jump up ^ Bring, Harry [@LLPOS] (December 14, 2017). "Today is Criminal Minds prep day 1 on episode 1316 "Last Gasp". Directed by: @_Adam_Rodriguez Written by :Stephanie SenGupta. Preps 2 days before our holiday hiatus and 5 days after Jan 3rd. Shoots mid-January" (Tweet). Retrieved December 14, 2017 – via Twitter.
Jump up ^ Bring, Harry [@LLPOS] (January 11, 2018). "Starting prep on 1317 today. "The Capilano's". Director: @GUBLERNATION Writer: @ecmess" (Tweet). Retrieved January 11, 2018 – via Twitter.
Jump up ^ Bring, Harry [@LLPOS] (September 26, 2017). "Yes. 1318" (Tweet). Retrieved November 1, 2017 – via Twitter.
Jump up ^ Frye, Kelly [@KellyFrye] (February 15, 2018). "Mr. & Mrs. Simmons with our fab writer & fab director @crimmindscbs we look like we're shooting Men In Black, but I promise it's Criminal Minds 🔪🧠" (Tweet). Retrieved February 15, 2018 – via Twitter.
Jump up ^ Bring, Harry [@LLPOS] (February 14, 2018). "Me and Diana Valentine @Ladydi2000 on Valentine's Day. Prep day 1 on episode 1320 "All You Can Eat" written by @KarenMaser" (Tweet). Retrieved February 14, 2018 – via Twitter.
Jump up ^ Bring, Harry [@LLPOS] (February 15, 2018). "Location scouting 1321. "Mixed Signals" Written by @e4meredith and @erikstiller Directed by Alec Smight" (Tweet). Retrieved February 15, 2018 – via Twitter.
Jump up ^ Porter, Rick (October 13, 2017). "'This Is Us' and 'The Good Doctor' score big in premiere week broadcast Live +7 ratings". TV by the Numbers. Retrieved October 13, 2017.
Jump up ^ Porter, Rick (October 18, 2017). "'The Orville,' 'Once Upon a Time,' 5 more shows double in week 2 broadcast Live +7 ratings". TV by the Numbers. Retrieved October 18, 2017.
Jump up ^ Porter, Rick (October 25, 2017). "'The Good Doctor' is the most-watched show on TV in week 3 broadcast Live +7 ratings". TV by the Numbers. Retrieved October 25, 2017.
Jump up ^ Porter, Rick (November 2, 2017). "'Chicago PD,' 10 more shows double in week 4 broadcast Live +7 ratings". TV by the Numbers. Retrieved November 2, 2017.
Jump up ^ Porter, Rick (November 9, 2017). "'Grey's Anatomy' keeps rolling along in week 5 broadcast Live +7 ratings". TV by the Numbers. Retrieved November 9, 2017.
Jump up ^ Porter, Rick (November 27, 2017). "'This Is Us' rides high in week 7 broadcast Live +7 ratings". TV by the Numbers. Retrieved November 27, 2017.
Jump up ^ Porter, Rick (December 5, 2017). "'Jane the Virgin,' 10 other shows double in week 8 broadcast Live +7 ratings". TV by the Numbers. Retrieved December 5, 2017.
Jump up ^ Porter, Rick (December 13, 2017). "'Chicago Med' premiere makes solid gains in week 9 broadcast Live +7 ratings". TV by the Numbers. Retrieved December 13, 2017.
Jump up ^ Porter, Rick (December 21, 2017). "'Agents of SHIELD' gets a good bump in week 11 broadcast Live +7 ratings". TV by the Numbers. Retrieved December 26, 2017.
Jump up ^ Otterson, Joe (January 22, 2018). "Delayed Viewing Ratings: 'Big Bang Theory' Winter Premiere Starts 2018 Strong". Variety. Retrieved February 1, 2018.
Jump up ^ Porter, Rick (January 26, 2017). "'The Brave' goes from low to … less low: Week 16 broadcast Live +7 ratings". TV by the Numbers. Retrieved January 26, 2017.
Jump up ^ Porter, Rick (February 1, 2018). "'This Is Us,' 'The Good Doctor' both double in week 17 broadcast Live +7 ratings". TV by the Numbers. Retrieved February 1, 2018.
Jump up ^ Porter, Rick (February 8, 2018). "'This Is Us' sets NBC records in week 18 broadcast Live +7 ratings". TV by the Numbers. Retrieved February 8, 2018.
Jump up ^ Porter, Rick (February 15, 2018). "Closing the book on the Super Bowl 'This Is Us': Week 19 broadcast Live +7 ratings". TV by the Numbers. Retrieved February 15, 2018.
External links[edit]
Criminal Minds on IMDb
Criminal Minds at TV.com
[hide]
v
t
e
Criminal Minds
Criminal Minds
Episodes
Season 1
2
3
4
5
6
7
8
9
10
11
12
13
Characters
Jason Gideon
Aaron Hotchner
Elle Greenaway
Derek Morgan
Spencer Reid
Jennifer Jareau
Penelope Garcia
Emily Prentiss
David Rossi
Alex Blake
Matt Simmons
Video game
Suspect Behavior
Sam Cooper
Penelope Garcia
Beyond Borders
Episodes
Season 1
2
Characters
Jack Garrett
Clara Seger
Russ Montgomery
Matt Simmons
Mae Jarvis
International
South Korean
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text:
Limit of a function
From Wikipedia, the free encyclopedia
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x
sin  x x {\displaystyle {\frac {\sin x}{x}}}
1 0.841471...
0.1 0.998334...
0.01 0.999983...
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1. In other words, the limit of (sin x)/x as x approaches zero equals 1.
Part of a series of articles about
Calculus
Fundamental theorem
Limits of functions
Continuity
Mean value theorem
Rolle's theorem
Differential[show]
Definitions
Derivative (generalizations)
Differential
infinitesimal
of a function
total
Concepts
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e
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Contents
[hide]
1 History
2 Motivation
3 Functions of a single variable
3.1 Existence and one-sided limits
3.2 More general subsets
3.3 Deleted versus non-deleted limits
3.4 Examples
3.4.1 Non-existence of one-sided limit(s)
3.4.2 Non-equality of one-sided limits
3.4.3 Limits at only one point
3.4.4 Limits at countably many points
4 Functions on metric spaces
5 Functions on topological spaces
6 Limits involving infinity
6.1 Limits at infinity
6.2 Infinite limits
6.3 Alternative notation
6.4 Limits at infinity for rational functions
7 Functions of more than one variable
8 Sequential limits
9 Other characterizations
9.1 In terms of sequences
9.2 In non-standard calculus
9.3 In terms of nearness
10 Relationship to continuity
11 Properties
11.1 Limits of compositions of functions
11.2 Limits of special interest
11.2.1 Rational functions
11.2.2 Trigonometric functions
11.2.3 Exponential functions
11.2.4 Logarithmic functions
11.3 L'Hôpital's rule
11.4 Summations and integrals
12 See also
13 Notes
14 References
History[edit]
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).
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).
Motivation[edit]
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.
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.
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.
The initial informal statement can now be explicated:
The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.
This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.
To say that
lim x → p f ( x ) = L , {\displaystyle \lim _{x\to p}f(x)=L,\,}
means that ƒ(x) can be made as close as desired to L by making x close enough, but not equal, to p.
The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.
Functions of a single variable[edit]
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
lim x → p f ( x ) = L , {\displaystyle \lim _{x\to p}f(x)=L,\ }
if the following property holds:
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.
The value of the limit does not depend on the value of f(p), nor even that p be in the domain of f.
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.
Here again the limit does not depend on f(p) being well-defined.
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.
Existence and one-sided limits[edit]
Main article: One-sided limit
The limit as: x → x0+ ≠ x → x0−. Therefore, the limit as x → x0 does not exist.
Alternatively x may approach p from above (right) or below (left), in which case the limits may be written as
lim x → p + f ( x ) = L {\displaystyle \lim _{x\to p^{+}}f(x)=L}
or
lim x → p − f ( x ) = L {\displaystyle \lim _{x\to p^{-}}f(x)=L}
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.
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.
If the limit does not exist then the oscillation of f at p is non-zero.
More general subsets[edit]
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.
This limit is often written
L = lim x → p x ∈ S f ( x ) . {\displaystyle L={\underset {x\in S}{\lim _{x\to p}}}f(x).}
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 )} ).
Deleted versus non-deleted limits[edit]
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:
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.
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)).
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.
Examples[edit]
Non-existence of one-sided limit(s)[edit]
The function without a limit, at an essential discontinuity
The function
f ( x ) = { sin  5 x − 1 for x < 1 0 for x = 1 0.1 x − 1 for x > 1 {\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {0.1}{x-1}}&{\text{ for }}x>1\end{cases}}}
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.
The function
f ( x ) = { 1 x rational 0 x irrational {\displaystyle f(x)={\begin{cases}1&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
(the Dirichlet function) has no limit at any x-coordinate.
Non-equality of one-sided limits[edit]
The function
f ( x ) = { 1 for x < 0 2 for x ≥ 0 {\displaystyle f(x)={\begin{cases}1&{\text{ for }}x<0\\2&{\text{ for }}x\geq 0\end{cases}}}
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).
Limits at only one point[edit]
The functions
f ( x ) = { x x rational 0 x irrational {\displaystyle f(x)={\begin{cases}x&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
and
f ( x ) = { | x | x rational 0 x irrational {\displaystyle f(x)={\begin{cases}|x|&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}
both have a limit at x = 0 and it equals 0.
Limits at countably many points[edit]
The function
f ( x ) = { sin  x x irrational 1 x rational {\displaystyle f(x)={\begin{cases}\sin x&x{\text{ irrational }}\\1&x{\text{ rational }}\end{cases}}}
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.
Functions on metric spaces[edit]
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
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
if the following property holds:
For every ε > 0, there exists a δ > 0 such that dB(f(x), L) < ε whenever 0 < dA(x, p) < δ.
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.
An alternative definition using the concept of neighbourhood is as follows:
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
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.
Functions on topological spaces[edit]
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
lim x → p f ( x ) = L {\displaystyle \lim _{x\to p}f(x)=L}
if the following property holds:
For every open neighborhood V of L, there exists an open neighborhood U of p such that f(U ∩ Ω − {p}) ⊆ V.
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 ".
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.
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.
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.
Limits involving infinity[edit]
Limits at infinity[edit]
The limit of this function at infinity exists.
For f(x) a real function, the limit of f as x approaches infinity is L, denoted
lim x → ∞ f ( x ) = L , {\displaystyle \lim _{x\to \infty }f(x)=L,}
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:
∀ ε > 0 ∃ c ∀ x > c : | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\;\exists c\;\forall x>c:\;|f(x)-L|<\varepsilon } .
Similarly, the limit of f as x approaches negative infinity is L, denoted
lim x → − ∞ f ( x ) = L , {\displaystyle \lim _{x\to -\infty }f(x)=L,}
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:
∀ ε > 0 ∃ c ∀ x < c : | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\;\exists c\;\forall x<c:\;|f(x)-L|<\varepsilon } .
For example
lim x → − ∞ e x = 0. {\displaystyle \lim _{x\to -\infty }e^{x}=0.\,}
Infinite limits[edit]
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
lim x → a f ( x ) = ∞ , {\displaystyle \lim _{x\to a}f(x)=\infty ,\,}
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 } .
These ideas can be combined in a natural way to produce definitions for different combinations, such as
lim x → ∞ f ( x ) = ∞ , lim x → a + f ( x ) = − ∞ . {\displaystyle \lim _{x\to \infty }f(x)=\infty ,\lim _{x\to a^{+}}f(x)=-\infty .\,}
For example
lim x → 0 + ln  x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty .\,}
Limits involving infinity are connected with the concept of asymptotes.
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
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
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.
Alternative notation[edit]
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):
lim x → 0 + 1 x = + ∞ , lim x → 0 − 1 x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}{1 \over x}=+\infty ,\lim _{x\to 0^{-}}{1 \over x}=-\infty .}
In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:
lim x → 0 + 1 x = lim x → 0 − 1 x = lim x → 0 1 x = ∞ . {\displaystyle \lim _{x\to 0^{+}}{1 \over x}=\lim _{x\to 0^{-}}{1 \over x}=\lim _{x\to 0}{1 \over x}=\infty .}
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.
Limits at infinity for rational functions[edit]
Horizontal asymptote about y = 4
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):
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.
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.
Functions of more than one variable[edit]
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,
lim ( x , y ) → ( p , q ) f ( x , y ) = L {\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=L}
if
for every ε > 0 there exists a δ > 0 such that for all (x,y) with 0 < ||(x,y) − (p,q)|| < δ, then |f(x,y) − L| < ε
where ||(x,y) − (p,q)|| represents the Euclidean distance. This can be extended to any number of variables.
Sequential limits[edit]
Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X and L ∈ Y.
The sequential limit of f as x tends to p is L if, for every sequence (xn) in X − {p} that converges to p, the sequence f(xn) converges to L.
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.
Other characterizations[edit]
In terms of sequences[edit]
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:
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
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.
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.
In non-standard calculus[edit]
In non-standard calculus the limit of a function is defined by:
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
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]
In terms of nearness[edit]
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
lim x → a f ( x ) = L {\displaystyle \lim _{x\to a}f(x)=L}
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.
Relationship to continuity[edit]
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:
lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}f(x)=f(c).}
(We have here assumed that c is a limit point of the domain of f.)
Properties[edit]
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.
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).
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.
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.
lim x → p ( f ( x ) + g ( x ) ) = lim x → p f ( x ) + lim x → p g ( x ) lim x → p ( f ( x ) − g ( x ) ) = lim x → p f ( x ) − lim x → p g ( x ) lim x → p ( f ( x ) ⋅ g ( x ) ) = lim x → p f ( x ) ⋅ lim x → p g ( x ) lim x → p ( f ( x ) / g ( x ) ) = lim x → p f ( x ) / lim x → p g ( x ) {\displaystyle {\begin{matrix}\lim \limits _{x\to p}&(f(x)+g(x))&=&\lim \limits _{x\to p}f(x)+\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)-g(x))&=&\lim \limits _{x\to p}f(x)-\lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)\cdot g(x))&=&\lim \limits _{x\to p}f(x)\cdot \lim \limits _{x\to p}g(x)\\\lim \limits _{x\to p}&(f(x)/g(x))&=&{\lim \limits _{x\to p}f(x)/\lim \limits _{x\to p}g(x)}\end{matrix}}}
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
q + ∞ = ∞ for q ≠ −∞
q × ∞ = ∞ if q > 0
q × ∞ = −∞ if q < 0
q / ∞ = 0 if q ≠ ± ∞
(see extended real number line).
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.
Limits of compositions of functions[edit]
In general, from knowing that
lim y → b f ( y ) = c {\displaystyle \lim _{y\to b}f(y)=c} and lim x → a g ( x ) = b {\displaystyle \lim _{x\to a}g(x)=b} ,
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:
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} ).
As an example of this phenomenon, consider the following functions that violates both additional restrictions:
f ( x ) = g ( x ) = { 0 if x ≠ 0 1 if x = 0 . {\displaystyle f(x)=g(x)={\begin{cases}0&{\text{if }}x\neq 0\\1&{\text{if }}x=0\end{cases}}.}
Since the value at f(0) is a removable discontinuity,
lim x → a f ( x ) = 0 {\displaystyle \lim _{x\to a}f(x)=0} for all a {\displaystyle a} .
Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that
f ( f ( x ) ) = { 1 if x ≠ 0 0 if x = 0 {\displaystyle f(f(x))={\begin{cases}1&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}
and so
lim x → a f ( f ( x ) ) = 1 {\displaystyle \lim _{x\to a}f(f(x))=1} for all a {\displaystyle a} .
Limits of special interest[edit]
Rational functions[edit]
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}} ,
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}}}}
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.
Trigonometric functions[edit]
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}
Exponential functions[edit]
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}
Logarithmic functions[edit]
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}}}
L'Hôpital's rule[edit]
Main article: l'Hôpital's rule
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:
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,
then:
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)}}}
Normally, the first condition is the most important one.
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}}.}
Summations and integrals[edit]
Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.
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.
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} .
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} .
See also[edit]
List of limits
One-sided limit
Limit of a sequence
Net (topology)
Big O notation
Limit superior and limit inferior
l'Hôpital's rule
Squeeze theorem
Non-standard calculus
Notes[edit]
Jump up ^ For example, "Limit" at Encyclopaedia of Mathematics
Jump up ^ Keisler, H. Jerome (2008), "Quantifiers in limits" (PDF), Andrzej Mostowski and foundational studies, IOS, Amsterdam, pp. 151–170
Jump up ^ Hrbacek, K. (2007), "Stratified Analysis?", in Van Den Berg, I.; Neves, V., The Strength of Nonstandard Analysis, Springer
Jump up ^ Bŀaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18 (1), arXiv:1202.4153  , doi:10.1007/s10699-012-9285-8
References[edit]
MacTutor History of Weierstrass.
MacTutor History of Bolzano
Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)
Apostol, Tom M., Mathematical Analysis, 2nd ed. Addison–Wesley, 1974. ISBN 0-201-00288-4.
Bartle, Robert (1967), The elements of real analysis, Wiley
Burton, David M. (1997), The History of Mathematics: An introduction (Third ed.), New York: McGraw–Hill, pp. 558–559, ISBN 0-07-009465-9
Felscher, Walter (2000), "Bolzano, Cauchy, Epsilon, Delta", American Mathematical Monthly, Mathematical Association of America, 107 (9): 844–862, doi:10.2307/2695743, JSTOR 2695743 .
Courant, Richard (1924), Vorlesungen über Differential- und Integralrechnung, Springer Verlag
Grabiner, Judith V. (1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus", American Mathematical Monthly, Mathematical Association of America, 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545, collected in Who Gave You the Epsilon? ISBN 978-0-88385-569-0 pp. 5–13 . Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf
Hardy, G.H. (1921), A course in pure mathematics, Cambridge University Press
Hubbard, John H. (2015), Vector calculus, linear algebra, and differential forms: A unified approach (Fifth ed.), Matrix Editions
Miller, Jeff (1 December 2004), Earliest Uses of Symbols of Calculus, retrieved 2008-12-18 .
Page, Warren; Hersh, Reuben; Selden, Annie; et al., eds. (2002), "Media Highlights", The College Mathematics, Mathematical Association of America, 33 (2): 147–154, JSTOR Journal 2687124 Journal .
Sutherland, W. A., Introduction to Metric and Topological Spaces. Oxford University Press, Oxford, 1975. ISBN 0-19-853161-3.
Rudin, Walter (1964), Principles of mathematical analysis, McGraw-Hill
Whittaker; Watson (1904), A course of modern analysis, Cambridge University Press
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values: f(x)=\begin{cases}|x| &  x \text{ rational } \\ 0 &  x \text{ irrational }\end{cases}

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values: \lim_{x \to \infty} f(x) = \infty, \lim_{x \to a^+}f(x) = -\infty. \,

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\begin{matrix}
\lim\limits_{x \to p} & (f(x) + g(x)) & = & \lim\limits_{x \to p} f(x) + \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x) - g(x)) & = & \lim\limits_{x \to p} f(x) - \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)\cdot g(x)) & = & \lim\limits_{x \to p} f(x) \cdot \lim\limits_{x \to p} g(x) \\
\lim\limits_{x \to p} & (f(x)/g(x)) & = & {\lim\limits_{x \to p} f(x) / \lim\limits_{x \to p} g(x)}
\end{matrix}

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

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values: \lim_{x \to 0} \frac{\ln(1+ax)}{bx} = \frac{a}{b}

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values: \lim_{x \to 0} \frac{\log_c(1+ax)}{bx} = \frac{a}{b\ln c}

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Indeterminate forms

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values: \lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,

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values: \lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty

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values: \lim_{x \to -\infty} \int_x^b f(t) \; dt

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