Upload
barnard-warner
View
220
Download
0
Embed Size (px)
Citation preview
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-2
12.2 Techniques For Calculating Limits
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-3
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-4
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-5
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-6
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-7
12.2 Techniques For Calculating Limits
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-8
12.2 Techniques For Calculating Limits
Rules for Limits (Continued)
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-9
12.2 Techniques For Calculating Limits
Rules for Limits (Continued)
9. Power rule For any real number k,
provided this limit exists.
lim[ ( )] lim ( )k
k
x a x af x f x
Copyright © 2011 Pearson Education, Inc. Slide 12.2-10
12.2 Techniques For Calculating Limits
Rules for Limits (Continued)
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-11
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-12
12.2 Finding a Limit of a Rational Function
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
Copyright © 2011 Pearson Education, Inc. Slide 12.2-13
12.2 Finding a Limit of a Rational Function
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