1 - Limits of Functions- Definition

8/6/2019 1 - Limits of Functions- Definition

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LIMITS

of

FUNCTIONS


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LIMITS OF FUNCTIONS

OBJECTIVES:

define limits;

illustrate limits and its theorems; andevaluate limits applying the given theorems.


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DEFINITION: Limits

The most basic use of limits is to describe how a

function behaves as the independent variableapproaches a given value. For example let us

examine the behavior of the function

for x-values closer and closer to 2. It is evident from

the graph and the table in the next slide that the

values of f(x) get closer and closer to 3 as the values

of x are selected closer and closer to 2 on either the

left or right side of 2. We describe this by sayingthat the limit of is 3 as x

approaches 2 from either side, we write

1xx)x(f 2 !

1xx)x(f 2 !

31xxlim2

2x !p


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2

3

f(x)

f(x)

x

y

1xxy 2 !

x 1.9 1.95 1.99 1.995 1.999 2 2.001 2.005 2.01 2.05 2.1

F(x) 2.71 2.852 2.97 2.985 2.997 3.003 3.015 3.031 3.152 3.31

left side right side


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1.1.1 (p. 70)Limits (An Informal View)

This leads us to the following general idea.


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EXAMPLE

Use numerical evidence to make a conjecture about

the value of

1x

1xlim

1x

p

Although the function , this has no

bearing on the limit.

The table shows sample x-values approaching 1 from

the left side and from the right side. In both cases the

corresponding values of f(x) appear to get closer andcloser to 2, and hence we conjecture that

and is consistent with the graph of f.

1x

1x)x(f

!

2

1x

1xli

1x!

p


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Figure 1.1.9 (p. 71)

x .99 .999 .9999 .99999 1 1.00001 1.0001 1.001 1.01

F(x) 1.9949 1.9995 1.99995 1.999995 2.000005 2.00005 2.0005 2.004915


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THEOREMS ON LIMITS

Our strategy for finding limits algebraically has two parts:

First we will obtain the limits of some simpler function

Then we will develop a list of theorems that will enable us

to use the limits of simple functions as building blocks for

finding limits of more complicated functions.


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We start with the following basic theorems, which are

illustrated in Fig 1.2.1

Theorem 1.2.1 (p. 80)

axkk !!pp axax

limblima

numbers.realbekandaLetTheorem1.2.1


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Figure 1.2.1 (p. 80)


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33li33li33li

exa ple,For

a.ofvaluesallforaxaskf(x)

whyexplainswhichvaries,xaskatfixedre ain

f(x)ofvaluesthethenfunction,constantaiskxfIf

x0x-25x!!!

pp

!

ppp T

Example 1.

TT

!!!

pp!

pppxxx

If

x-2x0xlim2lim0lim

example,For

.axfthattruebealsomustitathen xx,xf

Example 2.


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Theorem 1.2.2 (p. 81)

The following theorem will be our basic tool for finding limits

algebraically


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This theorem can be stated informally as follows:

a) The limit of a sum is the sum of the limits.

b) The limit of a difference is the difference of the limits.

c) The limits of a product is the product of the limits.

d) The limits of a quotient is the quotient of the limits,provided the limit of the denominator is not zero.

e) The limit of the nth root is the nth root of the limit.

A constant factor can be moved through a limit symbol.


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31

58

5)4(2

5limxlim2

5limx2lim5x2lim.1

4x4x

4x4x4x

!

!

!

!

!

pp

ppp

6

12-18

12)3(6

12limx6lim12x6lim.23x3x3x

!

!!

!ppp

13

131

2)3(534

2limxlim5xlim4lim

2limx5limxlim4lim

2x5limx4lim)2x5(x4lim.3

3x3x3x3x

3x3x3x3x

3x3x3x

!

!

!

!

!

!

pppp

pppp

ppp

EXAMPLE Evaluate the following limits.


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21

10

425

52

4lixli5

xli2

4lix5li

x2li

4x5

x2li.4

5x5x

5x

5x5x

5x

5x

!

!

!

!

pp

p

pp

p

p

3375

15633

6lixli3

6lix3li

6x3li6x3li.5

33

3

3x3x

3

3x3x

3

3x

3

3x

!

!!

!

!

!

pp

pp

pp

2

3

4

9

3x

1x8li

3x

1x8li.6

1x1x

!!

!

pp


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OR

When evaluating the limit of a function at a givenvalue, simply replace the variable by the indicated

limit then solve for the value of the function:

22

3

lim 3 4 1 3 3 4 3 1

27 12 1

38

x

x xp

!

!

!


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EXAMPLE Evaluate the following limits.

2x

8xlim.1

3

2x

p

Solution:

0

0

0

88

22

82

2x

8xli

33

2x!

!

!

p

2

2

2

2

2

3

2

2 2 4li

2

li 2 4

2 2 2 4

4 4 4

12

8li 122

x

x

x

x x x

x

x x

x

x

p

p

p

!

!

! !

!

@ !

Equivalent function:

(indeterminate)


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Note: In evaluating a limit of a quotient which

reduces to , simplify the fraction. Just removethe common factor in the numerator and

denominator which makes the quotient .

To do this use factoring or rationalizing thenumerator or denominator, wherever the radical is.

0

0

0

0


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0

2 2 2 2 0lim

0 0x

x

xp

! !

0 0

0

0

2 2 2 2 2 2lim lim

2 2 2 2

1 1 1 2

lim 42 2 2 2 2 2

2 2 2lim

4

x x

x

x

x x x

x x x x

x

x

x

p p

p

p

!

! ! !

@ !

x

22xlim.2

0x

p

Solution:

Rationalizing the numerator:

(indeterminate)


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9x4

27x8lim.3

2

3

0x

p

Solution:

Rationalizing the numerator:

(indeterminate)3

2

3

3

22

38 27

8 27 27 27 02lim

4 9 9 9 034 9

2

x

x

xp

! ! !

3 32 2

32

23

2

2

2 3 4 6 98 27li li

4 9 2 3 2 3

4 6 9 9 9 9li

2 3 3 3

27 9 3 3 2

6 2 22

x x

x

x x x x

x x x

x x

x

p p

p

!

! !

! ! ! !


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5x

3x2xli.4

2

3

2x

p

Solution:

33

222

2 2 2 32 3lim

5 2 5

8 4 34 5

15

9

15

3

x

x x

xp

!

!

!

!


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EXERCISES

5w4w

7w7wli10.

2x

8xli.5

19x9x2li9.4y

y8y4li.4

1y2y

3y2y1yli8.

1x

4x3xli.3

1x

3x2x3x2li7.

4x3x

1x2li.2

1x9

1x3li6.2x5x4li.1

2

2

1w

3

2x

2

134

5x

3

1

3

2y

2

2

1y3

2

1x

2

23

1x21x

2

3

1x

2

3x

pp

pp

pp

pp

pp

Evaluate the following limits.

1 - Limits of Functions- Definition

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