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# Lecture 2 - Limits and One-Sided Limits

Apr 04, 2018

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Keith Smith
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Limit of a Function and

One-sided limits

Mathematics 53

Institute of Mathematics (UP Diliman)

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For today

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

3 One-sided Limits

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For today

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

3 One-sided Limits

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Introduction

Given a function f(x) and a

R ,

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Introduction

Given a function f(x) and a

R ,

what is the value of f at x near a,

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Introduction

Given a function f(x) and a

R ,

what is the value of f at x near a,

but not equal to a?

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Illustration 1

Consider f(x) = 3x 1.

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 1

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 5

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

Based on the table, as x gets closer and closer to 1,

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Illustration 1

Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

Based on the table, as x gets closer and closer to 1, f(x) gets closer and closer

to 2.

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Ill i 1

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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4

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Ill t ti 1

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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4

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Ill t ti 1

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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4

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Ill stration 1

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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

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Illustration 1

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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

As x gets closer and closer to 1, f(x) gets closer and closer to 2.

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Illustration 2

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Illustration 2

Consider: g(x) =3x2 4x + 1

x 1

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Illustration 2

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Illustration 2

Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

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Illustration 2

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Illustration 2

Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 2

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Consider: g(x) =3x2 4x + 1

x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1, g(x) gets closer and closer to 2.

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1,

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Illustration 3

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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4

As x gets closer and closer to 1, h(x) gets closer and closer to 2.

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Limit

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Limit

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Intuitive Notion of a Limit

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Limit

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Intuitive Notion of a Limita R , L R

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Limit

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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a

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Limit

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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a

The limit of f(x) as x approaches a is L

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Limit

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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a

The limit off(x)

asx

approachesa

isL

if the values of f(x) get closer and closer to L as x assumes values getting closer

and closer to a but not reaching a.

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Limit

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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a

The limit off(x)

asx

approachesa

isL

if the values of f(x) get closer and closer to L as x assumes values getting closer

and closer to a but not reaching a.

Notation:limxa f(x) = L

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Examples

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f(x) = 3x 1

1 1 2 3

1

1

2

3

4

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Examples

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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1)

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Examples

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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1) = 2

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Examples

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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1) = 2

Note: In this case, limx1

f(x)

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Examples

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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1) = 2

Note: In this case, limx1

f(x) = f(1).

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Examples

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g(x) =3

x

2

4

x +1

x 1

1 1 2 31

1

2

3

4

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Examples

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g(x) =3

x

2

4

x+ 1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x + 1x 1

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Examples

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g(x) =3x2

4x + 1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x + 1x 1 = 2

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Examples

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g(x) =3x2

4x + 1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x + 1x 1 = 2

Note: Though g(1) is undefined,limx1

g(x) exists.

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Examples

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h(x) =

3x

1, x

= 1

0, x = 1

1 1 2 31

1

2

3

4

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Examples

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h(x) =

3x

1, x

= 1

0, x = 1

1 1 2 31

1

2

3

4

limx1

h(x)

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Examples

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h(x) =

3x

1, x

= 1

0, x = 1

1 1 2 31

1

2

3

4

limx1

h(x) = 2

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Examples

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h(x) =

3x

1, x

= 1

0, x = 1

1 1 2 31

1

2

3

4

limx1

h(x) = 2

Note: h(1) = limx1 h(x).

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Some Remarks

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Remark

In finding limxa f(x):

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Some Remarks

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Remark

In finding limxa f(x):

We only need to consider values of x very close to a but not exactly at a.

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Some Remarks

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Remark

In finding limxa f(x):

We only need to consider values of x very close to a but not exactly at a.

Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).

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Some Remarks

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Remark

In finding limxa f(x):

We only need to consider values of x very close to a but not exactly at a.

Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).

We let x approach a from BOTH SIDES of a.

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Some Remarks

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Iff(x)

does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

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Some Remarks

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Iff(x)

does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

Some Remarks

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

76/290

If f(x) does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

3 2 1 1 2 3

1

2

3

0

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

77/290

Some Remarks

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

78/290

If f(x) does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

Some Remarks

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

79/290

If f(x) does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1?

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

Some Remarks

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

80/290

If f(x) does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

Some Remarks

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

81/290

If f(x) does not approach any

particular real number as x

approaches a, then we say

limxa f(x) does not exist (dne).

e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)

3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.

limx0

H(x) dne

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40

For today

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

82/290

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

3 One-sided Limits

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 16 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

83/290

Theorem

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

84/290

Theorem

If limxa f(x) exists, then it is unique.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

85/290

Theorem

If limxa f(x) exists, then it is unique.

If c R , then limxa c

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

86/290

Theorem

If limxa f(x) exists, then it is unique.

If c R , then limxa c = c.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

87/290

Theorem

If limxa f(x) exists, then it is unique.

If c R , then limxa c = c.limxa x

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

88/290

Theorem

If limxa f(x) exists, then it is unique.

If c R , then limxa c = c.limxa x = a

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

89/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

90/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)]

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

91/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

92/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

93/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

94/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)]

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

95/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)] = limxa

f(x) limxa

g(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

96/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40

Limit Theorems

Th

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

97/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] =

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40

Limit Theorems

Th

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

98/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] = c limxa f(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40

Limit Theorems

Th

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

99/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] = c limxa f(x) = cL1

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

100/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

101/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

102/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

103/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

104/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

105/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n =

limxa f(x)

n

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Limit Theorems

Theorem

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

106/290

Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .

limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2

limxa[c f(x)] = c limxa f(x) = cL1

limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n =

limxa f(x)

n = (L1)n

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40

Evaluate: limx1

(2x2 + 3x 4)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

107/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

108/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

109/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 +

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

110/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

111/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

112/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

113/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

114/290

= 2

limx1 x2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

115/290

= 2

limx1 x2

+ 3

limx1 x

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

116/290

= 2

limx1 x2

+ 3

limx1 x limx1 4

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

117/290

= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

118/290

= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

119/290

= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

120/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

121/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

122/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

l2

l l

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

123/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

124/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4

= 5

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

125/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4

= 5

In general:

Remark

If f is a polynomial function, then limxa f(x)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

126/290

= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4

= 5

In general:

Remark

If f is a polynomial function, then limxa f(x) = f(a).

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

127/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3 + 3x2 x + 1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

128/290

limx2

4x3 + 3x2 x + 1x2 + 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3

+ 3x2

x + 1lim

x 2(4x3 + 3x2 x + 1)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

129/290

limx2

4x + 3x x + 1x2 + 2

= x2 lim

x2(x2 + 2)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3

+ 3x2

x + 1lim

x2(4x3 + 3x2 x + 1)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

130/290

limx2

4x + 3x x + 1x2 + 2

= x2 lim

x2(x2 + 2)

=4(8)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1lim

x2(4x3 + 3x2 x + 1)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

131/290

limx2

4x + 3x x + 1x2 + 2

= x 2 lim

x2(x2 + 2)

=4(8) + 3(4)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)

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limx2

4x + 3x x + 1x2 + 2

= x 2 lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)

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limx2

4x + 3x x + 1x2 + 2

= x 2lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)

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limx2

4x + 3x x + 1x2 + 2

=lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2

= 176

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)

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limx2

4x + 3x x + 1x2 + 2

=lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2

= 176

Remark

If f is a rational function and f(a) is defined,

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 12

=lim

x2(4x3 + 3x2

x + 1)

2

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limx2

4x + 3x x + 1x2 + 2 lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2

= 176

Remark

If f is a rational function and f(a) is defined, then lim

xaf(x) = f(a).

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxan

f(x) =n

limxa f(x),

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f ( )

f ( ),

provided limxa f(x) > 0 when n is even.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxan

f(x) =n

limxa f(x),

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f ( )

f ( ),

provided limxa f(x) > 0 when n is even.

limx3

3x

1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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f ( )

f ( )

provided limxa f(x) > 0 when n is even.

limx3

3x

1 = limx3

(3x

1)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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f ( )

f ( )

provided limxa f(x) > 0 when n is even.

limx3

3x

1 = limx3

(3x

1) =

8

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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f ( )

f ( )

provided limxa f(x) > 0 when n is even.

limx3

3x

1 = limx3

(3x

1) =

8 = 2

2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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( )

( )

provided limxa f(x) > 0 when n is even.

limx3

3x

1 = limx3

(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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provided lim

xa f(x) > 0 when n is even.

limx3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) =

nlimxa f(x)

,

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provided lim

xa f(x) > 0 when n is even.

limx3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf

(x

) = nlimxa f

(x

),

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provided lim

xa f(x) > 0 when n is even.

limx3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) = nlimxa

f(x),

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provided lim

xa f(x) > 0 when n is even.

lim

x3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

limx7/2

4

3 2x

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) = nlimxa

f(x),

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provided lim

xa f(x) > 0 when n is even.

lim

x3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

limx7/2

4

3 2x dne

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) = nlimxa

f(x),

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provided lim

xa f(x) > 0 when n is even.

lim

x3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) = nlimxa

f(x),

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provided lim

xa f(x) > 0 when n is even.

lim

x3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4 =??

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Theorem

Suppose limxa f(x) exists and n N . Then,

limxa

nf(x) = nlimxa

f(x),

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provided lim

xa f(x) > 0 when n is even.

lim

x3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4 =?? (for now)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

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151/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

152/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

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=

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

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=

limx3

2x2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

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=

limx3

2x2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

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=

limx3

2x2

limx3

(5x + 1)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

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=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3

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=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3

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=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3

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=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4

28

3

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3

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=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4

28

3

=1

8

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40

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162/290

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

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163/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

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x1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0

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x1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1)

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166/290

x1

x1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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x1

x1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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x x

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.

Since we are just taking the limit as x 1,

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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170/290

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.

Since we are just taking the limit as x 1,

limx1

3x2 4x + 1x 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.

Since we are just taking the limit as x 1,

limx1

3x2 4x + 1x 1 = limx1(3x 1)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Consider: g(x) =3x2 4x + 1

x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?

Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.

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But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.

Since we are just taking the limit as x 1,

limx1

3x2 4x + 1x 1 = limx1(3x 1) = 2.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40

Definition

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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0

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174/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

lim

xa

f(x)

g(x)

is called an indeterminate form of type0

0.

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175/290

yp0

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

lim

xa

f(x)

g(x)

is called an indeterminate form of type0

0.

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yp0

Remarks:

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

limx

a

f(x)

g(x)

is called an indeterminate form of type0

0.

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0

Remarks:

1 If f(a) = 0 and g(a) = 0, then f(a)g(a)

is undefined!

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

limx

a

f(x)

g(x)

is called an indeterminate form of type0

0.

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0

Remarks:

1 If f(a) = 0 and g(a) = 0, then f(a)g(a)

is undefined!

2 The limit above MAY or MAY NOT exist.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

limx

a

f(x)

g(x)

is called an indeterminate form of type0

0.

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0

Remarks:

1 If f(a) = 0 and g(a) = 0, then f(a)g(a)

is undefined!

2 The limit above MAY or MAY NOT exist.

3 Some techniques used in evaluating such limits are:

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Definition

If limxa f(x) = 0 and limxag(x) = 0 then

limx

a

f(x)

g(x)

is called an indeterminate form of type0

0.

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180/290

0

Remarks:

1 If f(a) = 0 and g(a) = 0, then f(a)g(a)

is undefined!

2 The limit above MAY or MAY NOT exist.

3 Some techniques used in evaluating such limits are:

FactoringRationalization

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1

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181/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1

0

0

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182/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1

0

0

limx 1

x2 + 2x + 1

x + 1

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183/290

x1 x + 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x 1x2 + 2x + 1

x + 1= lim

x 1

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184/290

x1 x + 1 x1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

185/290

x 1 x + 1 x 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

x + 1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

186/290

x 1 x + 1 x 1 x + 1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

x + 1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

187/290

x 1 x + 1 x 1 x + 1

= limx

1(x + 1)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

188/290

Examples

Evaluate: limx

1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

x + 1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

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= limx

1(x + 1)

= (1 + 1)

= 0

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

190/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

191/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

192/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

193/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2(x + 2)(x2 2x + 4)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

194/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

195/290

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2(x + 2)(x2 2x + 4)

(x + 2)(x 2)2 2 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

196/290

= limx

2

x2 2x + 4x

2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2(x + 2)(x2 2x + 4)

(x + 2)(x 2)2 2 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

197/290

= limx

2

x2 2x + 4x

2

=4 + 4 + 4

2 2

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2(x + 2)(x2 2x + 4)

(x + 2)(x 2)2 2 + 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

198/290

= limx

2

x2 2x + 4x

2

=4 + 4 + 4

2 2= 3

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40

Examples

Evaluate: limx4

x2 162x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

199/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

200/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

201/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

202/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

203/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

204/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

205/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

206/290

= limx4

(x 4)(x + 4)(2 +x)4 x

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

207/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

( )( )(

)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

208/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]

= (8)(4)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

( )( )(

)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

209/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]

= (8)(4)

= 32

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

210/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

211/290

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

212/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

213/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

214/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

215/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

216/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

217/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

218/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8(x 8)(x + 1)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

219/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8(x 8)(x + 1)( 3

x2 + 2 3

x + 4)

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

220/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8(x 8)(x + 1)( 3

x2 + 2 3

x + 4)

1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

221/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8(x 8)(x + 1)( 3

x2 + 2 3

x + 4)

1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

222/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)=

1

9(4 + 4 + 4)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+ 23

x + 43x2 + 2 3

x + 4

= limx8

x 8(x 8)(x + 1)( 3

x2 + 2 3

x + 4)

1

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

223/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)=

1

9(4 + 4 + 4)

=1

108

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40

For today

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

224/290

3 One-sided Limits

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

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225/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1.

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

226/290

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3

1

2

3

4

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

227/290

3

2

1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3

1

1

2

3

4

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

228/290

3

2

1

As x approaches 1 through values less than 1,

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3

1

1

2

3

4

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

229/290

3

2

1

As x approaches 1 through values less than 1, f(x) approaches 2.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3

1

1

2

3

4

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

230/290

3

2

1

As x approaches 1 through values less than 1, f(x) approaches 2.As x approaches 1 through values greater than 1,

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3

1

1

2

3

4

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

231/290

3

2

1

As x approaches 1 through values less than 1, f(x) approaches 2.As x approaches 1 through values greater than 1, f(x) approaches 1.

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40

Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

• 7/29/2019 Lecture 2 - Limits and One-Sided Limits

232/290

1

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 40

Illustration 5

Consider: g(x) =

x

2 1 1 2

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