Lesson 2: The Concept of Limit

Section 1.3
The Concept of Limit

V63.0121.002.2010Su, Calculus I

New York University

May 18, 2010

Announcements

WebAssign Class Key: nyu 0127 7953
Office Hours: MR 5:00–5:45, TW 7:50–8:30, CIWW 102 (here)
Quiz 1 Thursday on 1.1–1.4

. . . . . .

Announcements

WebAssign Class Key: nyu
0127 7953
Office Hours: MR
5:00–5:45, TW 7:50–8:30,
CIWW 102 (here)
Quiz 1 Thursday on
1.1–1.4

. . . . . .

V63.0121.002.2010Su, Calculus I (NYU) Section 1.3 The Concept of Limit May 18, 2010 2 / 32

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.

. . . . . .


Last Time

Key concept: function
Properties of functions: domain and range
Kinds of functions: linear, polynomial, power, rational, algebraic,
transcendental.

. . . . . .


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 to decide if a limit lim f(x) exists.
x→a
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 given
error of L. The idea is that points x within the tolerance level of
a are taken by f to y-values within the error level of L, with the
possible exception of a itself.
If Player 1 can do this, he wins the round. If he cannot, he
loses the game: the limit cannot be L.
Step 4 Player 2 go back to Step 2 with a smaller error. Or, he can give
up and concede that 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
.

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


Outline

Heuristics


Examples

Pathologies


. . . . . .


Playing the Error-Tolerance game with x2
Example
Find lim x2 if it exists.
x→0

. . . . . .


Example
x→0

Solution

Step 1 Player 1: I claim the limit is zero.

. . . . . .


Example
x→0

Solution

Step 2 Player 2: I challenge you to make x2 within 0.01 of 0.

. . . . . .


Example
x→0

Solution

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.

. . . . . .


Example
x→0

Solution

Step 4 Player 2: OK, smart guy. Can you make x2 within 0.0001 of 0?

. . . . . .


Example
x→0

Solution

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. …

. . . . . .


Example
x→0

Solution

Can you convince Player 2 that Player 1 can win every round?

. . . . . .


Example
x→0

Solution

Can you convince Player 2 that Player 1 can win every round? Yes, by
setting the tolerance equal to the square root of the error, Player 1 can
always win. Player 2 should give up and concede that the limit is 0. . . . . . .


Graphical version of the E-T game with x2

. .
y

. . .
x
.
.

. . . . . .


Graphical version of the E-T game with x2

. .
y

. . .
x
.
.

No matter how small an error band Player 2 picks, Player 1 can
find a fitting tolerance band.
. . . . . .


Limit of a piecewise function

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?

. . . . . .


The E-T game with a piecewise function

y
.
.

. . .
1

. . ..
x

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
I
. think the limit is 1
. . ..
x

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
I
. . ..
x
C
. an you fit an error of 0.5?

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
H
. ow about this for a tolerance?
. . ..
x

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
H
. . ..
x
.
No. Part of
graph inside
. 1.
−
blue is not inside
green
.
.
. . . . . .



y
.
.

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



y
.
.

. . .
1
. think the limit is −1
I
. . ..
x

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
. think the limit is −1
I
. . ..
x
C

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
H
. . ..
x
C

. 1.
−

.
.
. . . . . .



y
.
.
.
No. Part of
. graph inside
. .
1
blue is not inside
. ow about this for a tolerance? green
H
. . ..
x

. 1.
−

.
.
. . . . . .



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

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
I
. . ..
x

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
I
. . ..
x
C

. 1.
−

.
.
. . . . . .



y
.
.

. . .
1
H
. . ..
x
.
No. None of
. 1.
−
graph inside blue
is inside green

.
.
. . . . . .



y
.
.

. . .
1

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

.
.
. . . . . .



y
.
.

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

. 1.
−

.
.
. . . . . .


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.

. . . . . .


The error-tolerance game on the right
y
.

. .
1

. x
.

. 1.
−

. . . . . .


y
.

. .
1

. x
.

.
All of graph in-
. 1.
− side blue is in-
side green

. . . . . .


y
.

.
All of graph in- . .
1
side blue is in-
side green
. x
.

. 1.
−

. . . . . .


y
.

.
All of graph in- . .
1
side blue is in-
side 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−

. . . . . .


Another Example

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

. . . . . .


The error-tolerance game with lim (1/x)
x→0
y
.

.?.
L

. x
.
0
.

. . . . . .


x→0
y
.

.
The graph escapes
the green, so no good

.?.
L

. x
.
0
.

. . . . . .


x→0
y
.

E
. ven worse!

.?.
L

. x
.
0
.

. . . . . .


x→0
y
.

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

.?.
L

. x
.
0
.

. . . . . .


Another (Bad) Example: Unboundedness

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

Solution
The limit does not exist because the function is unbounded near 0.
Later we will talk about 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

y
.
Fundamental Concept: . .
1
limit
Error-Tolerance game . x
.
gives a methods of arguing
limits do or do not exist
Limit FAIL: jumps,
. 1.
−
unboundedness, sin(π/x)

FAIL
.

. . . . . .


Lesson 2: The Concept of Limit

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