Lesson 3: Limits

Section 1.3The Concept of Limit

V63.0121.041, Calculus I

New York University

September 13, 2010

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I First written HW due WednesdayI Get-to-know-you survey and photo deadline is October 1

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. . . . . .

Announcements

I Let us know if you boughta WebAssign license lastyear and cannot login

I First written HW dueWednesday

I Get-to-know-you surveyand photo deadline isOctober 1

V63.0121.041, Calculus I (NYU) Section 1.3 The Concept of Limit September 13, 2010 2 / 36

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Guidelines for written homework

I Papers should be neat and legible. (Use scratch paper.)I Label with name, lecture number (041), recitation number, date,

assignment number, book sections.I Explain your work and your reasoning in your own words. Use

complete English sentences.


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Rubric

Points Description of Work3 Work is completely accurate and essentially perfect.

Work is thoroughly developed, neat, and easy to read.Complete sentences are used.

2 Work is good, but incompletely developed, hard to read,unexplained, or jumbled. Answers which are not ex-plained, even if correct, will generally receive 2 points.Work contains “right idea” but is flawed.

1 Work is sketchy. There is some correct work, but most ofwork is incorrect.

0 Work minimal or non-existent. Solution is completely in-correct.


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Examples of written homework: Don't


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Examples of written homework: Do


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Examples of written homework: DoWritten Explanations


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Examples of written homework: DoGraphs


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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.


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)


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Outline

Heuristics

Errors and tolerances

Examples

Pathologies

Precise Definition of a Limit


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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.


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Outline

Heuristics


Examples

Pathologies



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The error-tolerance game

A game between two players (Dana and Emerson) to decide if a limitlimx→a

f(x) exists.

Step 1 Dana proposes L to be the limit.Step 2 Emerson challenges with an “error” level around L.Step 3 Dana chooses a “tolerance” level around a so that points x

within that tolerance of a (not counting a itself) are taken tovalues y within the error level of L. If Dana cannot, Emersonwins and the limit cannot be L.

Step 4 If Dana’s move is a good one, Emerson can challenge again orgive up. If Emerson gives up, Dana wins and the limit is L.


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.

.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 Even if Emerson shrinks the error, Dana can still move.


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.

.This tolerance is too big

.Still too big.This looks good.So does this

.a

.L




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.


.a

.L




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.

.This tolerance is too big

.Still too big

.This looks good.So does this

.a

.L




. . . . . .


.


.a

.L




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.

.This tolerance is too big.Still too big

.This looks good

.So does this

.a

.L




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.

.This tolerance is too big.Still too big.This looks good

.So does this

.a

.L




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.


.a

.L




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Outline

Heuristics


Examples

Pathologies



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Example

Find limx→0

x2 if it exists.

Solution

I Dana claims the limit is zero.I If Emerson challenges with an error level of 0.01, Dana needs to

guarantee that −0.01 < x2 < 0.01 for all x sufficiently close tozero.

I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.I If Emerson re-challenges with an error level of 0.0001 = 10−4,

what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒

∣∣∣x2∣∣∣ < 10−4.I Dana has a shortcut: By setting tolerance equal to the square root

of the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.


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Example

Find limx→0

x2 if it exists.

Solution

I Dana claims the limit is zero.

I If Emerson challenges with an error level of 0.01, Dana needs toguarantee that −0.01 < x2 < 0.01 for all x sufficiently close tozero.






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Example

Find limx→0

x2 if it exists.

Solution








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Example

Find limx→0

x2 if it exists.

Solution



I If −0.1 < x < 0.1, then 0 ≤ x2 < 0.01, so Dana wins the round.

I If Emerson re-challenges with an error level of 0.0001 = 10−4,what should Dana’s tolerance be? A tolerance of 0.01 worksbecause |x| < 10−2 =⇒




. . . . . .

Example

Find limx→0

x2 if it exists.

Solution




what should Dana’s tolerance be?

A tolerance of 0.01 worksbecause |x| < 10−2 =⇒




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Example

Find limx→0

x2 if it exists.

Solution





∣∣∣x2∣∣∣ < 10−4.

I Dana has a shortcut: By setting tolerance equal to the square rootof the error, Dana can win every round. Once Emerson realizesthis, Emerson must give up.


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Example

Find limx→0

x2 if it exists.

Solution








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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


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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


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. .x

.y

..−1

..1 .

.

.Part of graphinside blue is notinside green


I These are the only good choices; the limit does not exist.


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. .x

.y

..−1

..1 .

..Part of graphinside blue is notinside green




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. .x

.y

..−1

..1 .

.





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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.


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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.


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. .x

.y

..−1

..1 .

.

.Part of graphinside blue isinside green


I So limx→0+

f(x) = 1 and limx→0−

f(x) = −1


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. .x

.y

..−1

..1 .

..Part of graphinside blue isinside green


I So limx→0+


f(x) = −1


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. .x

.y

..−1

..1 .

.



I So limx→0+


f(x) = −1


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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


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Example

Find limx→0+

1xif it exists.

SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that

limx→0+

1x= +∞


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. .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


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. .x

.y

.0

..L?

.The graph escapesthe green, so no good

.Even worse!



. . . . . .


. .x

.y

.0

..L?




. . . . . .


. .x

.y

.0

..L?

.The graph escapesthe green, so no good

.Even worse!



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. .x

.y

.0

..L?




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Example

Find limx→0+

1xif it exists.

SolutionThe limit does not exist because the function is unbounded near 0.Next week we will understand the statement that

limx→0+

1x= +∞


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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


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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


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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


24k− 1

for any integer k


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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


24k− 1

for any integer k


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Weird, wild stuff

Example

Find limx→0

sin(πx

)if it exists.


I f(x) = 1 when x =2

4k+ 1for any integer k


24k− 1

for any integer k


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Weird, wild stuff

Example

Find limx→0

sin(πx

)if it exists.


I f(x) = 1 when x =2

4k+ 1for any integer k

I f(x) = −1 when x =2

4k− 1for any integer k


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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.


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Outline

Heuristics


Examples

Pathologies



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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


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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


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Outline

Heuristics


Examples

Pathologies



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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.


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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


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Summary: Many perspectives on limits

I Graphical: L is the value the function “wants to go to” near aI Heuristical: f(x) can be made arbitrarily close to L by taking x

sufficiently close to a.I Informal: the error/tolerance gameI Precise: if for every ε > 0 there is a corresponding δ > 0 such that

if 0 < |x− a| < δ, then |f(x)− L| < ε.I Algebraic/Formulaic: next time


Lesson 3: Limits

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