Lesson 3: The Limit of a Function (slides)

Sec on 1.3
The Limit of a Func on
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University

January 31, 2011

Announcements
First wri en HW due Wednesday February 2
. Get-to-know-you survey and photo deadline is February 11

Announcements

First wri en HW due
Wednesday February 2
Get-to-know-you survey
and photo deadline is
February 11

Guidelines for written homework

Papers should be neat and legible. (Use scratch paper.)
Label with name, lecture number (011), recita on number,
date, assignment number, book sec ons.
Explain your work and your reasoning in your own words. Use
complete English sentences.

Rubric
Points Descrip on of Work
3 Work is completely accurate and essen ally 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 explained, 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. Solu on is completely
incorrect.

Written homework: Do
Written Explanations

Objectives
Understand and state the
informal deﬁni on of a
limit.
Observe limits on a
graph.
Guess limits by algebraic
manipula on.
Guess limits by numerical
informa on.

Yoda on teaching course concepts
You must unlearn
what you have
learned.
In other words, we are
building up concepts and
allowing ourselves only to
speak in terms of what we
personally have produced.

Zeno’s Paradox

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

Outline

Heuris cs

Errors and tolerances

Examples

Precise Deﬁni on of a Limit

Heuristic Definition of a Limit
Defini on
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.

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.

lim f(x) exists.
x→a
Step 2 Emerson challenges with an “error” level around L.

lim f(x) exists.
x→a
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not coun ng a itself) are taken to
values y within the error level of L. If Dana cannot, Emerson
wins and the limit cannot be L.

lim f(x) exists.
x→a
Step 3 Dana chooses a “tolerance” level around a so that points x
within that tolerance of a (not coun ng 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 (ver cal) strip
must also be inside the green (horizontal) strip.

This tolerance is too big
L

.
a

S ll too big
L

.
a

This looks good
L

.
a

So does this
L

.
a


L

.
a
Even if Emerson shrinks the error, Dana can s ll move.

Playing the E-T Game
Example
Describe how the the Error-Tolerance game would be played to
determine lim x2 .
x→0

Solu on

Example
determine lim x2 .
x→0

Solu on
Dana claims the limit is zero.

Example
determine lim x2 .
x→0

Solu on
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
determine lim x2 .
x→0

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

Example
determine lim x2 .
x→0

Solu on

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

Example
determine lim x2 .
x→0

Solu on

If Emerson re-challenges with an error level of 0.0001 = 10−4 ,
what should Dana’s tolerance be?
A tolerance of 0.01 works because
|x| < 10−2 =⇒ x2 < 10−4 .

Example
determine lim x2 .
x→0

Solu on
Dana has a shortcut: By se ng tolerance equal to the square
root of the error, Dana can win every round. Once Emerson
realizes this, Emerson must give up.

Graphical version of E-T game
with x2
y

.
x

Graphical version of E-T game
with x2
y

No ma er how small an
error Emerson picks,
Dana can ﬁnd a ﬁ ng
tolerance band.

.
x

A piecewise-deﬁned function
Example
|x|
Find lim if it exists.
x→0 x

Example
|x|
x→0 x

Solu on
The func on can also be wri en as
{
|x| 1 if x > 0;
=
x −1 if x < 0

What would be the limit?

The E-T game with a piecewise
function
|x|
x→0 x y

1

. x

−1

function
|x|
x→0 x y

1
I think the limit is 1
. x

−1

function
|x|
x→0 x y

1
I think the limit is 1
. x
Can you ﬁt an error of 0.5?

−1

function
|x|
x→0 x y

1
How about
this for a tol- . x
erance?
−1

function
|x|
x→0 x y

1
How about
this for a tol- .
No. Part of x
erance? graph inside
−1 blue is not
inside green

function
|x|
x→0 x y

Oh, I guess 1
the limit isn’t
1 .
No. Part of x
graph inside
−1 blue is not
inside green

function
|x|
x→0 x y

1
I think the limit
is −1 . x

−1

function
|x|
x→0 x y

1
I think the limit
is −1 . Can you ﬁt xan
error of 0.5?
−1

function
|x|
x→0 x y

1
How about
. Can you ﬁt xan
this for a tol-
error of 0.5?
erance?
−1

function
|x|
x→0 x y
No. Part of
graph inside
1 blue is not
How about inside green
this for a tol- . x
erance?
−1

function
|x|
x→0 x y
No. Part of
graph inside
Oh, I guess 1 blue is not
the limit isn’t inside green
−1 . x

−1

function
|x|
x→0 x y

1
I think the limit
is 0 . x

−1

function
|x|
x→0 x y

1
I think the limit
is 0 . Can you ﬁt xan
error of 0.5?
−1

function
|x|
x→0 x y

1
How about
this for a tol- . No. None of x
erance? graph inside
−1 blue is inside
green

function
|x|
x→0 x y

1
Oh, I guess
the limit isn’t . No. None of x
0 graph inside
−1 blue is inside
green

function
|x|
x→0 x y

1
I give up! I
guess there’s . x
no limit!
−1

One-sided limits
Deﬁni on
We write
lim f(x) = L
x→a+
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 as
we like) by taking x to be suﬃciently close to a and greater than a.

One-sided limits
Deﬁni on
We write
lim f(x) = L
x→a−
and say

“the limit of f(x), as x approaches a from the le , 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 and less than a.

|x| |x|
Find lim+ and lim− if they exist.
x→0 x x→0 x
y

1

. x

−1

|x| |x|
x→0 x x→0 x
y

1

. x

Part of graph
−1 inside blue is
inside green

|x| |x|
x→0 x x→0 x
y

Part of graph 1
inside blue is
inside green
. x

−1

Example
|x|
x→0 x

Solu on
The error-tolerance game fails, but

lim f(x) = 1 lim f(x) = −1
x→0+ x→0−

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

The error-tolerance game with 1/x
y

1
Find lim+ if it exists. L?
x→0 x

. x
0

y

The graph escapes
the green, so no
good
1
x→0 x

. x
0

y

Even worse!

1
x→0 x

. x
0

y

The limit does not exist
because the func on is
unbounded near 0
1
x→0 x

. x
0

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

Solu on
The limit does not exist because the func on 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)
π/2

π . 0

3π/2

Function values
x π/x sin(π/x)
π 0 π/2

π . 0

3π/2

Function values
x π/x sin(π/x)
π 0 π/2
2π 0

π . 0

3π/2

Function values
x π/x sin(π/x)
π 0 π/2
2π 0
kπ 0

π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
2π 0
kπ 0

π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
kπ 0

π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0

π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0
9π/2 1

3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
π/2 1
5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
9π/2 1
((4k + 1)π)/2 1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
1/2 2π 0
1/k kπ 0
2 π/2 1
2/5 5π/2 1 π . 0
2/9 9π/2 1
((4k + 1)π)/2 1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
7π/2 −1 3π/2

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
3π/2 −1
7π/2 −1 3π/2
((4k − 1)π)/2 −1

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
7π/2 −1 3π/2
((4k − 1)π)/2 −1

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
2/7 7π/2 −1 3π/2
((4k − 1)π)/2 −1

Function values
x π/x sin(π/x)
1 π 0 π/2
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/(4k + 1) ((4k + 1)π)/2 1
2/3 3π/2 −1
2/7 7π/2 −1 3π/2
2/(4k − 1) ((4k − 1)π)/2 −1

Weird, wild stuﬀ
Example
(π )
x→0 x

Solu on

f(x) = 0 when x =

f(x) = 1 when x =

f(x) = −1 when x =

Weird, wild stuﬀ
Example
(π )
x→0 x

Solu on
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

Solu on
1
k
2
4k + 1
f(x) = −1 when x =

Weird, wild stuﬀ
Example
(π )
x→0 x

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

Graph
Here is a graph of the func on:
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.

What could go wrong?
Summary of Limit Pathologies

How could a func on fail to have a limit? Some possibili es:
le - and right- hand limits exist but are not equal
The func on is unbounded near a
Oscilla on with increasingly high frequency near a

Meet the Mathematician
Augustin Louis Cauchy
French, 1789–1857
Royalist and Catholic
made contribu ons in geometry,
calculus, complex analysis,
number theory
created the deﬁni on of limit
we use today but didn’t
understand it

Precise Deﬁnition of a Limit
No, this is not going to be on the test
Let f be a func on 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
L−ε

.
a


L+ε
L
L−ε

.
a−δ a a+δ


This δ is too big
L+ε
L
L−ε

.
a−δ a a+δ


This δ looks good
L+ε
L
L−ε

.
a−δ a a+δ


So does this δ
L+ε
L
L−ε

.
a−δ aa+δ

Summary
Many perspectives on limits
Graphical: L is the value the func on
“wants to go to” near a y
Heuris cal: f(x) can be made arbitrarily 1
close to L by taking x suﬃciently close
to a. . x
Informal: the error/tolerance game
Precise: if for every ε > 0 there is a
−1
corresponding δ > 0 such that if
0 < |x − a| < δ, then |f(x) − L| < ε.
Algebraic: next me
FAIL

Lesson 3: The Limit of a Function (slides)

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