Limits at Infinity

Section 3.5

1

Horizontal Asymptotes

The line y = L is a horizontal asymptote of the graph of f if

x

lim  

L or x



f x



L

2

Example

 3

x

 1 2

x

 5

x

lim  3

x

 1 2

x

 5 

x

lim  3

x

 1 2

x

 5  Check out TABLE on calculator.

3

Limits at Infinity

If r is a positive rational number and c is any real

x

lim 

x c r

 0 Furthermore, if x r

c

0 .

x

lim 

x r

 is defined for x < 0, then 4

Examples

1) Find :

x

lim 3   2

x

2) Find :

x

lim  4

x

 5 3

x

 1 Strategy: Divide each term of the numerator and denominator by the highest power of x in the denominator. 5

Examples

5

x

 3 3) Find : lim

x



x

2  1 4) Find :

x

lim  10

x

2 2

x

2   1 3 6

Examples

2

x

3 5) Find :

x

lim 

x

 7  2 7

Limits at Infinity

Guidelines for finding limits at infinity of rational functions

If degree of numerator is less than degree of denominator, the limit is zero.

If degree of numerator is equal to degree of denominator, the limit is the ratio of the leading coefficients.

If degree of numerator is greater than degree of denominator, the limit does not exist.

8

End Behavior

End behavior of the graph of functions: Example: Describe end behavior of 

x

2  3

x

9

End Behavior

Example: Describe end behavior of  2

x

2

x

  1

x

10

Homework

Section 3.5 page 205 #1, 3-8 all, 17, 21-25 odd, Full curve sketches for 55, 59, 65, 67 11