Lesson 3: Limits

Section 1.3
The Concept of Limit

V63.0121.041, Calculus I

New York University

September 13, 2010

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

Announcements

Let us know if you bought
a WebAssign license last
year and cannot login
First written HW due
Wednesday
Get-to-know-you survey
and photo deadline is
October 1

. . . . . .

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

Guidelines for written homework

Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (041), recitation number, date,
assignment number, book sections.
Explain your work and your reasoning in your own words. Use
complete English sentences.

. . . . . .


Rubric

Points Description of Work
3 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 of
work is incorrect.
0 Work minimal or non-existent. Solution is completely in-
correct.

. . . . . .


Examples of written homework: Don't

. . . . . .


Examples of written homework: Do

. . . . . .


Written Explanations

. . . . . .


Graphs

. . . . . .


Objectives

Understand and state the
informal definition of a limit.
Observe limits on a graph.
Guess limits by algebraic
manipulation.
Guess limits by numerical
information.

. . . . . .


Zeno's Paradox

That which is in
locomotion must arrive
at the half-way stage
before it arrives at the
goal.
(Aristotle Physics VI:9, 239b10)

. . . . . .


Outline

Heuristics

Errors and tolerances

Examples

Pathologies

Precise Definition of a Limit

. . . . . .


Heuristic Definition of a Limit

Definition
We write
lim f(x) = L
x→a

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 as
we like) by taking x to be sufficiently close to a (on either side of a) but
not equal to a.

. . . . . .


Outline

Heuristics


Examples

Pathologies


. . . . . .


The error-tolerance game

A game between two players (Dana and Emerson) to decide if a limit
lim f(x) exists.
x→a
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 to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.
Step 4 If Dana’s move is a good one, Emerson can challenge again or
give up. If Emerson gives up, Dana wins and the limit is L.

. . . . . .



L
.

.
a
.

. . . . . .



L
.

.
a
.

To be legit, the part of the graph inside the blue (vertical) strip
must also be inside the green (horizontal) strip.

. . . . . .



T
. his tolerance is too big

L
.

.
a
.


. . . . . .



S
. till too big

L
.

.
a
.


. . . . . .



T
. his looks good

L
.

.
a
.


. . . . . .



S
. o does this

L
.

.
a
.


. . . . . .



L
.

.
a
.

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


Outline

Heuristics


Examples

Pathologies


. . . . . .


Example
Find lim x2 if it exists.
x→0

. . . . . .


Example
x→0

Solution

Dana claims the limit is zero.

. . . . . .


Example
x→0

Solution

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 to
zero.

. . . . . .


Example
x→0

Solution

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

. . . . . .


Example
x→0

Solution

zero.
If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?

. . . . . .


Example
x→0

Solution

zero.
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .

. . . . . .


Example
x→0

Solution

zero.
what should Dana’s tolerance be? A tolerance of 0.01 works
because |x| < 10−2 =⇒ x2 < 10−4 .
Dana has a shortcut: By setting tolerance equal to the square root
of the error, Dana can win every round. Once Emerson realizes
this, Emerson must give up.
. . . . . .


Example
|x|
Find lim if it exists.
x→0 x

. . . . . .


Example
|x|
x→0 x

Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0

What would be the limit?

. . . . . .



y
.

. .
1

. x
.

. 1.
−

. . . . . .



y
.

. .
1

. x
.

.
Part of graph
. 1.
− inside blue is not
inside green

. . . . . .



y
.

.
Part of graph
inside blue is not . .
1
inside green

. x
.

. 1.
−

. . . . . .



y
.

.
Part of graph
inside blue is not . .
1
inside green

. x
.

. 1.
−

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


One-sided limits

Definition
We write
lim f(x) = L
x→a+

and say

“the limit of f(x), as x approaches a from the right, equals L”

we like) by taking x to be sufficiently close to a and greater than a.

. . . . . .


One-sided limits

Definition
We write
lim f(x) = L
x→a−

and say

“the limit of f(x), as x approaches a from the left, equals L”

we like) by taking x to be sufficiently close to a and less than a.

. . . . . .


y
.

. .
1

. x
.

. 1.
−

. . . . . .


y
.

. .
1

. x
.

.
Part of graph
. 1.
− inside blue is
inside green

. . . . . .


y
.

.
Part of graph . .
1
inside blue is
inside green
. x
.

. 1.
−

. . . . . .


y
.

.
Part of graph . .
1
inside blue is
inside green
. x
.

. 1.
−

So lim+ f(x) = 1 and lim f(x) = −1
x→0 x→0−
. . . . . .


Example
|x|
x→0 x

Solution
The function can also be written as
{
|x| 1 if x > 0;
=
x −1 if x < 0

What would be the limit?
The error-tolerance game fails, but

lim f(x) = 1 lim f(x) = −1
x→0+ x→0−

. . . . . .


Example
1
Find lim+ if it exists.
x→0 x

. . . . . .


y
.

.?.
L

. x
.
0
.

. . . . . .


y
.

.
The graph escapes
the green, so no good

.?.
L

. x
.
0
.

. . . . . .


y
.

E
. ven worse!

.?.
L

. x
.
0
.

. . . . . .


y
.

.
The limit does not ex-
ist because the func-
tion is unbounded near
0

.?.
L

. x
.
0
.

. . . . . .


Example
1
Find lim+ if it exists.
x→0 x

Solution
The limit does not exist because the function is unbounded near 0.
Next week we will understand the statement that
1
lim+ = +∞
x→0 x

. . . . . .


Weird, wild stuff

Example
(π )
Find lim sin if it exists.
x→0 x

. . . . . .


Function values

x π/x sin(π/x) . /2
π
.
1 π 0
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 . .
π . ..
0
2/9 9π/2 1
2/13 13π/2 1
2/3 3π/2 −1
2/7 7π/2 −1 .
2/11 11π/2 −1 3
. π/2

. . . . . .


Weird, wild stuff

Example
(π )
x→0 x

. . . . . .


Weird, wild stuff

Example
(π )
x→0 x

f(x) = 0 when x =

f(x) = 1 when x =

f(x) = −1 when x =

. . . . . .


Weird, wild stuff

Example
(π )
x→0 x

1
f(x) = 0 when x = for any integer k
k
f(x) = 1 when x =

f(x) = −1 when x =

. . . . . .


Weird, wild stuff

Example
(π )
x→0 x

1
k
2
4k + 1
f(x) = −1 when x =

. . . . . .


Weird, wild stuff

Example
(π )
x→0 x

1
k
2
4k + 1
2
f(x) = −1 when x = for any integer k
4k − 1

. . . . . .


Weird, wild stuff continued

Here is a graph of the function:
y
.
. .
1

. x
.

. 1.
−

There are infinitely many points arbitrarily close to zero where f(x) is 0,
or 1, or −1. So the limit cannot exist.

. . . . . .


Outline

Heuristics


Examples

Pathologies


. . . . . .


What could go wrong?
Summary of Limit Pathologies

How could a function fail to have a limit? Some possibilities:
left- and right- hand limits exist but are not equal
The function is unbounded near a
Oscillation with increasingly high frequency near a

. . . . . .


Meet the Mathematician: Augustin Louis Cauchy

French, 1789–1857
Royalist and Catholic
made contributions in
geometry, calculus,
complex analysis, number
theory
created the definition of
limit we use today but
didn’t understand it

. . . . . .


Outline

Heuristics


Examples

Pathologies


. . . . . .


No, this is not going to be on the test

Let f be a function defined on an some open interval that contains the
number a, except possibly at a itself. Then we say that the limit of f(x)
as x approaches a is L, and we write

lim f(x) = L,
x→a

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
.

.
a
.

. . . . . .



L
. +ε
L
.
. −ε
L

.
a
.

. . . . . .



L
. +ε
L
.
. −ε
L

.
. − δ. . + δ
a aa

. . . . . .



T
. his δ is too big
L
. +ε
L
.
. −ε
L

.
. − δ. . + δ
a aa

. . . . . .



L
. +ε
L
.
. −ε
L

.
. −. δ δ
a . a+
a

. . . . . .



T
. his δ looks good
L
. +ε
L
.
. −ε
L

.
. −. δ δ
a . a+
a

. . . . . .



S
. o does this δ
L
. +ε
L
.
. −ε
L

.
. .− δ δ
aa .+
a

. . . . . .


Summary: Many perspectives on limits

Graphical: L is the value the function “wants to go to” near a
Heuristical: f(x) can be made arbitrarily close to L by taking x
sufficiently close to a.
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a corresponding δ > 0 such that
if 0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic/Formulaic: next time

. . . . . .


Lesson 3: Limits

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