Lesson 3: The Limit of a Function

Section 1.3The Limit of a Function

V63.0121, Calculus I

January 26–27, 2009

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Limit

Zeno’s Paradox

That which is inlocomotion mustarrive at thehalf-way stagebefore it arrives atthe goal.

(Aristotle Physics VI:9,239b10)

Outline

The Concept of LimitHeuristicsErrors and tolerancesExamplesPathologies

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

The error-tolerance game

A game between two players to decide if a limit limx→a

f (x) exists.

I Player 1: Choose L to be the limit.

I Player 2: Propose an “error” level around L.

I Player 1: Choose a “tolerance” level around a so that x-pointswithin that tolerance level are taken to y -values within theerror level.

If Player 1 can always win, limx→a

f (x) = L.


This tolerance is too bigStill too bigThis looks goodSo does this

a

L

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

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



a

L





a

L




This tolerance is too big

Still too bigThis looks goodSo does this

a

L





a

L




This tolerance is too big

Still too big

This looks goodSo does this

a

L





a

L




This tolerance is too bigStill too big

This looks good

So does this

a

L




This tolerance is too bigStill too bigThis looks good

So does this

a

L





a

L





a

L



Example

Find limx→0

x2 if it exists.

SolutionBy setting tolerance equal to the square root of the error, we canguarantee to be within any error.

Example

Find limx→0

x2 if it exists.

SolutionBy setting tolerance equal to the square root of the error, we canguarantee to be within any error.

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

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


x

y

−1

1

Part of graph in-side blue is notinside green


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


x

y

−1

1





x

y

−1

1





x

y

−1

1





x

y

−1

1





x

y

−1

1





x

y

−1

1





x

y

−1

1





x

y

−1

1




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 toL as we like) by taking x to be sufficiently close to a (on either sideof 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 toL as we like) by taking x to be sufficiently close to a (on either sideof a) and less than a.

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

Example

Find limx→0+

1

xif it exists.

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

limx→0+

1

x= +∞


x

y

0

L?

The graph escapes thegreen, so no good

Even worse!The limit does not existbecause the function isunbounded near 0


x

y

0

L?




x

y

0

L?




x

y

0

L?




x

y

0

L?




x

y

0

L?


Even worse!

The limit does not existbecause the function isunbounded near 0


x

y

0

L?


Even worse!

The limit does not existbecause the function isunbounded near 0

Example

Find limx→0+

1

xif it exists.

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

limx→0+

1

x= +∞

Example

Find limx→0

sin(π

x

)if it exists.

x

y

−1

1

What could go wrong?

How could a function fail to have a limit? Some possibilities:

I left- and right- hand limits exist but are not equal

I The function is unbounded near a

I Oscillation with increasingly high frequency near a

Meet the Mathematician: Augustin Louis Cauchy

I French, 1789–1857

I Royalist and Catholic

I made contributions ingeometry, calculus,complex analysis,number theory

I created the definition oflimit we use today butdidn’t understand it

Precise Definition of a Limit

Let f be a function defined on an some open interval that containsthe number a, except possibly at a itself. Then we say that thelimit 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− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L


L + ε

L− ε

a− δ a + δ

This δ is too big

a− δa + δ

This δ looks good

a− δa + δ

So does this δ

a

L

Lesson 3: The Limit of a Function

Education

example x

errortolerance gamela

errortolerance gameso

errortolerance game

errortolerance game

errortolerance gamey1

limit lim f x

values of f x