Lesson 3: The limit of a function

Section 2.2The Limit of a Function

Math 1a

September 28, 2007

Zeno’s Paradox

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

(Aristotle Physics VI:9,239b10)

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.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

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The error-tolerance game

This tolerance is too bigStill too bigThis looks goodSo does this

a

L

The error-tolerance game

This tolerance is too bigStill too bigThis looks goodSo does this

a

L

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page3of9

The error-tolerance game

This tolerance is too bigStill too bigThis looks goodSo does this

a

L

The error-tolerance game

This tolerance is too big

Still too bigThis looks goodSo does this

a

L

The error-tolerance game

This tolerance is too bigStill too bigThis looks goodSo does this

a

L

The error-tolerance game

This tolerance is too big

Still too big

This looks goodSo does this

a

L

The error-tolerance game

This tolerance is too bigStill too bigThis looks goodSo does this

a

L

The error-tolerance game

This tolerance is too bigStill too big

This looks good

So does this

a

L

The error-tolerance game

This tolerance is too bigStill too bigThis looks good

So does this

a

L

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page4of9

Examples

Example

Find limx→0

x2 if it exists.

Example

Find limx→0

|x |x

if it exists.

Example

Find limx→0+

1

xif it exists.

Example

Find limx→0

sin(π

x

)if it exists.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page5of9

Examples

Example

Find limx→0

x2 if it exists.

Example

Find limx→0

|x |x

if it exists.

Example

Find limx→0+

1

xif it exists.

Example

Find limx→0

sin(π

x

)if it exists.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page6of9

Examples

Example

Find limx→0

x2 if it exists.

Example

Find limx→0

|x |x

if it exists.

Example

Find limx→0+

1

xif it exists.

Example

Find limx→0

sin(π

x

)if it exists.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page7of9

Examples

Example

Find limx→0

x2 if it exists.

Example

Find limx→0

|x |x

if it exists.

Example

Find limx→0+

1

xif it exists.

Example

Find limx→0

sin(π

x

)if it exists.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page8of9

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

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.

Math 1a - September 28, 2007.GWBFriday, Sep 28, 2007

Page9of9

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

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

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

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

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

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

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