Copyright © 2011 Pearson Education, Inc. Slide 12.2-1 12.2 Techniques For Calculating Limits Rules...

Copyright © 2011 Pearson Education, Inc. Slide 12.2-1

12.2 Techniques For Calculating Limits

Rules for Limits

1. Constant rule If k is a constant real number,

2. Limit of x rule For the following rules, we assume that and

both exist

3. Sum and difference rules

lim .x ak k

lim .x ax a

lim[ ( ) ( )] lim ( ) lim ( ).x a x a x a

f x g x f x g x

lim ( )x a

f x

lim ( )x ag x



Rules for Limits

4. Product Rule

5. Quotient Rule

provided

lim[ ( ) ( )] lim ( ) lim ( ).x a x a x a

f x g x f x g x

lim ( )( )lim .

( ) lim ( )x a

x ax a

f xf x

g x g x

lim ( ) 0.x ag x


12.2 Finding a Limit of a Linear Function

Example Find

Solution

Rules 1 and 4

Rules 1 and 2

4lim (3 2 ).x

x

4 4 4lim (3 2 ) lim 3 lim 2x x x

x x

4 43 lim 2 lim

x xx

3 2 4

11


12.2 Finding a Limit of a Polynomial Function with One Term

Example Find

Solution Rule 4

Rule 1

Rule 4

Rule 2

2

5lim 3 .x

x

2 2

5 5 5lim 3 lim 3 limx x x

x x

2

53 lim

xx

5 53 lim lim

x xx x

3 5 5

75


12.2 Finding a Limit of a Polynomial Function with One Term

For any polynomial function in the form ( ) ,nf x kx

lim ( ) ( ).n

x af x k a f a


12.2 Finding a Limit of a Polynomial Function

Example Find .

Solution

Rule 3

3

2lim (4 6 1)x

x x

3 3

2 2 2 2lim (4 6 1) lim 4 lim 6 lim 1x x x x

x x x x

34 2 6 2 1

21



Rules for Limits (Continued)

For the following rules, we assume that and

both exist.

6. Polynomial rule If p(x) defines a polynomial function, then

lim ( )x a

f x

lim ( )x ag x

lim ( ) ( ).x ap x p a




7. Rational function rule If f(x) defines a rational

function with then

• Equal functions rule If f(x) = g(x) for all , then

lim ( ) ( ).x af x f a

( )

( )

p x

q x( ) 0q a

x a

lim ( ) lim ( ).x a x af x g x




9. Power rule For any real number k,

provided this limit exists.

lim[ ( )] lim ( )k

k

x a x af x f x




10. Exponent rule For any real number b > 0,

11. Logarithm rule For any real number b > 0 with ,

provided that

lim ( )( )lim .x af xf x

x ab b

1b

lim log ( ) log lim ( )b bx a x a

f x f x

lim ( ) 0.x af x


12.2 Finding a Limit of a Rational Function

Example Find

Solution Rule 7 cannot be applied directly since the denominator is 0. First factor the numerator and denominator

2

21

2 3lim .

3 2x

x x

x x

2

2

2 3 ( 3)( 1) 3

3 2 ( 2)( 1) 2

x x x x x

x x x x x



Solution Now apply Rule 8 with

and

so that f(x) = g(x) for all .

2

2

2 3( )

3 2

x xf x

x x

3

( )2

xg x

x

1x



Solution Rule 8

Rule 6

2

21 1

2 3 3lim lim

3 2 2x x

x x x

x x x

1 3

1 2

4

Copyright © 2011 Pearson Education, Inc. Slide 12.2-1 12.2 Techniques For Calculating Limits Rules...

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