Lesson 2: The Concept of Limit

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

. . . . . .

. . . . . .

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.


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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.


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.


. . . . . .

Outline

Heuristics


Examples

Pathologies



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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.


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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 If Player 2 shrinks the error, Player 1 can still win.


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.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




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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




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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




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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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.a

.L




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Outline

Heuristics


Examples

Pathologies



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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.


. . . . . .


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,





. . . . . .


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




. . . . . .


Example

Find limx→0

x2 if it exists.

Solution


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




. . . . . .


Example

Find limx→0

x2 if it exists.

Solution


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. …



. . . . . .


Example

Find limx→0

x2 if it exists.

Solution






. . . . . .


Example

Find limx→0

x2 if it exists.

Solution




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.


. . . . . .


Example

Find limx→0

x2 if it exists.

Solution






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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.


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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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.. ..x

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..y



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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


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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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The E-T game with a piecewise function

.. ..x

.

..y

..−1

..1 .

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.

.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


.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!


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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.Oh, I guess the limit isn’t 1

.I think the limit is −1







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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.Oh, I guess the limit isn’t −1.I think the limit is 0





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.. ..x

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..y

..−1

..1 .

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.







.No. Part ofgraph insideblue is not insidegreen.Oh, I guess the limit isn’t −1






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.. ..x

.

..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.No. None ofgraph inside blueis inside green




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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..y

..−1

..1 .

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.. ..x

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..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.


. . . . . .

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.


. . . . . .

The error-tolerance game on the right

. .x

.y

..−1

..1 .

.

.All of graph in-side blue is in-side green


I So limx→0+

f(x) = 1 and limx→0−

f(x) = −1


. . . . . .


. .x

.y

..−1

..1 .

.



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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. .x

.y

..−1

..1 .

.



I So limx→0+


f(x) = −1


. . . . . .


. .x

.y

..−1

..1 .

..All of graph in-side blue is in-side 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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. .x

.y

..−1

..1 .

.



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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. .x

.y

..−1

..1 .

.



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


. . . . . .

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= +∞


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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


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(1/x)

. .x

.y

.0

..L?




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(1/x)

. .x

.y

.0

..L?




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(1/x)

. .x

.y

.0

..L?

.The graph escapesthe green, so no good

.Even worse!



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(1/x)

. .x

.y

.0

..L?




. . . . . .


(1/x)

. .x

.y

.0

..L?

.The graph escapesthe green, so no good

.Even worse!



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(1/x)

. .x

.y

.0

..L?




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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= +∞


. . . . . .

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


. . . . . .

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


. . . . . .

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


. . . . . .

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


. . . . . .

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


. . . . . .

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


. . . . . .

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



. . . . . .

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.


. . . . . .

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


. . . . . .


.

.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


. . . . . .


.

.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


. . . . . .


.

.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


. . . . . .


.

.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


. . . . . .


.

.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


. . . . . .


.

.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


. . . . . .

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


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Lesson 2: The Concept of Limit

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