lesson 2: the concept of limit
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![Page 1: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/1.jpg)
Section 1.3The Concept of Limit
V63.0121.002.2010Su, Calculus I
New York University
May 18, 2010
Announcements
I WebAssign Class Key: nyu 0127 7953I Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here)I Quiz 1 Thursday on 1.1–1.4
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. . . . . .
Announcements
I WebAssign Class Key: nyu0127 7953
I Office Hours: MR5:00–5:45, TW 7:50–8:30,CIWW 102 (here)
I Quiz 1 Thursday on1.1–1.4
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 2 / 32
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. . . . . .
Objectives
I Understand and state theinformal definition of a limit.
I Observe limits on a graph.I Guess limits by algebraic
manipulation.I Guess limits by numerical
information.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 3 / 32
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. . . . . .
Last Time
I Key concept: functionI Properties of functions: domain and rangeI Kinds of functions: linear, polynomial, power, rational, algebraic,
transcendental.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 4 / 32
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Limit
. . . . . .
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. . . . . .
Zeno's Paradox
That which is inlocomotion must arriveat the half-way stagebefore it arrives at thegoal.
(Aristotle Physics VI:9, 239b10)
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 5 / 32
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. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 6 / 32
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. . . . . .
Heuristic Definition of a Limit
DefinitionWe write
limx→a
f(x) = L
and say
“the limit of f(x), as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L aswe like) by taking x to be sufficiently close to a (on either side of a) butnot equal to a.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 7 / 32
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. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 8 / 32
![Page 10: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/10.jpg)
. . . . . .
The error-tolerance game
A game between two players to decide if a limit limx→a
f(x) exists.
Step 1 Player 1 proposes L to be the limit.Step 2 Player 2 chooses an “error” level around L: the maximum
amount f(x) can be away from L.Step 3 Player 1 looks for a “tolerance” level around a: the maximum
amount x can be from a while ensuring f(x) is within the givenerror of L. The idea is that points x within the tolerance level ofa are taken by f to y-values within the error level of L, with thepossible exception of a itself.If Player 1 can do this, he wins the round. If he cannot, heloses the game: the limit cannot be L.
Step 4 Player 2 go back to Step 2 with a smaller error. Or, he can giveup and concede that the limit is L.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 9 / 32
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. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 12: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/12.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 13: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/13.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 14: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/14.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big
.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 15: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/15.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 16: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/16.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big
.Still too big
.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 17: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/17.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 18: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/18.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big
.This looks good
.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 19: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/19.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good
.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 20: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/20.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 21: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/21.jpg)
. . . . . .
The error-tolerance game
.
.This tolerance is too big.Still too big.This looks good.So does this
.a
.L
I To be legit, the part of the graph inside the blue (vertical) stripmust also be inside the green (horizontal) strip.
I If Player 2 shrinks the error, Player 1 can still win.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 10 / 32
![Page 22: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/22.jpg)
. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 11 / 32
![Page 23: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/23.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 24: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/24.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.
Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 25: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/25.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.
Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 26: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/26.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.
Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 27: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/27.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?
Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, soa tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 28: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/28.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 29: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/29.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round?
Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 30: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/30.jpg)
. . . . . .
Playing the Error-Tolerance game with x2
Example
Find limx→0
x2 if it exists.
Solution
Step 1 Player 1: I claim the limit is zero.Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.Step 3 Player 1: That’s easy. If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01,
so a tolerance of 0.1 fits your error of 0.01.Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?Step 5 Player 1: Sure. If −0.01 < x < 0.01, then 0 ≤ x2 < 0.0001, so
a tolerance of 0.01 fits your error of 0.0001. …
Can you convince Player 2 that Player 1 can win every round? Yes, bysetting the tolerance equal to the square root of the error, Player 1 canalways win. Player 2 should give up and concede that the limit is 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 12 / 32
![Page 31: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/31.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 32: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/32.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 33: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/33.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 34: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/34.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 35: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/35.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 36: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/36.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 37: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/37.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 38: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/38.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 39: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/39.jpg)
. . . . . .
Graphical version of the E-T game with x2
.. ..x
.
..y
I No matter how small an error band Player 2 picks, Player 1 canfind a fitting tolerance band.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 13 / 32
![Page 40: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/40.jpg)
. . . . . .
Limit of a piecewise function
Example
Find limx→0
|x|x
if it exists.
Solution
The function can also be written as
|x|x
=
{1 if x > 0;−1 if x < 0
What would be the limit?The error-tolerance game fails, but
limx→0+
f(x) = 1 limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
![Page 41: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/41.jpg)
. . . . . .
Limit of a piecewise function
Example
Find limx→0
|x|x
if it exists.
SolutionThe function can also be written as
|x|x
=
{1 if x > 0;−1 if x < 0
What would be the limit?
The error-tolerance game fails, but
limx→0+
f(x) = 1 limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 14 / 32
![Page 42: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/42.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 43: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/43.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 44: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/44.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 45: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/45.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 46: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/46.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 47: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/47.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1
.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 48: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/48.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1
.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 49: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/49.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1
.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 50: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/50.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 51: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/51.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 52: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/52.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1
.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 53: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/53.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1
.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 54: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/54.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1
.I think the limit is 0
.Can you fit an error of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 55: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/55.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 56: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/56.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 57: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/57.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?
.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 58: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/58.jpg)
. . . . . .
The E-T game with a piecewise function
.. ..x
.
..y
..−1
..1 .
.
.
.
.I think the limit is 1
.Can you fit an error of 0.5?
.How about this for a tolerance?
.No. Part ofgraph insideblue is not insidegreen
.Oh, I guess the limit isn’t 1.I think the limit is −1
.Can you fit an error of 0.5?
.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1.I think the limit is 0
.Can you fit an error of 0.5?.No. None ofgraph inside blueis inside green
.Oh, I guess thelimit isn’t 0
.I give up! Iguess there’sno limit!
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 15 / 32
![Page 59: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/59.jpg)
. . . . . .
One-sided limits
DefinitionWe write
limx→a+
f(x) = L
and say
“the limit of f(x), as x approaches a from the right, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L aswe like) by taking x to be sufficiently close to a and greater than a.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
![Page 60: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/60.jpg)
. . . . . .
One-sided limits
DefinitionWe write
limx→a−
f(x) = L
and say
“the limit of f(x), as x approaches a from the left, equals L”
if we can make the values of f(x) arbitrarily close to L (as close to L aswe like) by taking x to be sufficiently close to a and less than a.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 16 / 32
![Page 61: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/61.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 62: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/62.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 63: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/63.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 64: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/64.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 65: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/65.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
..All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 66: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/66.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 67: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/67.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 68: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/68.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
![Page 69: Lesson 2: The Concept of Limit](https://reader034.vdocument.in/reader034/viewer/2022051609/546248d1b4af9f5d1c8b473b/html5/thumbnails/69.jpg)
. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
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. . . . . .
The error-tolerance game on the right
. .x
.y
..−1
..1 .
.
.All of graph in-side blue is in-side green
.All of graph in-side blue is in-side green
I So limx→0+
f(x) = 1 and limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 17 / 32
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. . . . . .
Limit of a piecewise function
Example
Find limx→0
|x|x
if it exists.
SolutionThe function can also be written as
|x|x
=
{1 if x > 0;−1 if x < 0
What would be the limit?The error-tolerance game fails, but
limx→0+
f(x) = 1 limx→0−
f(x) = −1
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 18 / 32
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. . . . . .
Another Example
Example
Find limx→0+
1xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Later we will talk about the statement that
limx→0+
1x= +∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 19 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good
.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good
.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
The error-tolerance game with limx→0
(1/x)
. .x
.y
.0
..L?
.The graph escapesthe green, so no good.Even worse!
.The limit does not ex-ist because the func-tion is unbounded near0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 20 / 32
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. . . . . .
Another (Bad) Example: Unboundedness
Example
Find limx→0+
1xif it exists.
SolutionThe limit does not exist because the function is unbounded near 0.Later we will talk about the statement that
limx→0+
1x= +∞
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 21 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 22 / 32
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. . . . . .
Function values
x π/x sin(π/x)1 π 0
1/2 2π 01/k kπ 02 π/2 1
2/5 5π/2 12/9 9π/2 12/13 13π/2 12/3 3π/2 −12/7 7π/2 −12/11 11π/2 −1
.
..π/2
..π
..3π/2
. .0
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 23 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =
1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =1kfor any integer k
I f(x) = 1 when x =
24k+ 1
for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =1kfor any integer k
I f(x) = 1 when x =2
4k+ 1for any integer k
I f(x) = −1 when x =
24k− 1
for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
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. . . . . .
Weird, wild stuff
Example
Find limx→0
sin(πx
)if it exists.
I f(x) = 0 when x =1kfor any integer k
I f(x) = 1 when x =2
4k+ 1for any integer k
I f(x) = −1 when x =2
4k− 1for any integer k
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 24 / 32
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. . . . . .
Weird, wild stuff continued
Here is a graph of the function:
. .x
.y
..−1
..1
There are infinitely many points arbitrarily close to zero where f(x) is 0,or 1, or −1. So the limit cannot exist.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 25 / 32
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. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 26 / 32
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. . . . . .
What could go wrong?Summary of Limit Pathologies
How could a function fail to have a limit? Some possibilities:I left- and right- hand limits exist but are not equalI The function is unbounded near aI Oscillation with increasingly high frequency near a
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 27 / 32
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. . . . . .
Meet the Mathematician: Augustin Louis Cauchy
I French, 1789–1857I Royalist and CatholicI made contributions in
geometry, calculus,complex analysis, numbertheory
I created the definition oflimit we use today butdidn’t understand it
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 28 / 32
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. . . . . .
Outline
Heuristics
Errors and tolerances
Examples
Pathologies
Precise Definition of a Limit
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 29 / 32
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. . . . . .
Precise Definition of a LimitNo, this is not going to be on the test
Let f be a function defined on an some open interval that contains thenumber a, except possibly at a itself. Then we say that the limit of f(x)as x approaches a is L, and we write
limx→a
f(x) = L,
if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x− a| < δ, then |f(x)− L| < ε.
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 30 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
The error-tolerance game = ε, δ
.
.L+ ε
.L− ε
.a− δ .a+ δ
.This δ is too big
.a− δ.a+ δ
.This δ looks good
.a− δ.a+ δ
.So does this δ
.a
.L
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 31 / 32
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. . . . . .
Summary
I Fundamental Concept:limit
I Error-Tolerance gamegives a methods of arguinglimits do or do not exist
I Limit FAIL: jumps,unboundedness, sin(π/x)
. .x
.y
..−1
..1
.FAIL
V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 32 / 32