Limits and Horizontal Asymptotes:

1
Chapter 3

• Limits and Horizontal Asymptotes
• Limits at Infinity Revisited

2
Limits at Infinity

• This is a topic that we already covered, but we
  will look at it from a different perspective.
• When we determine limits at infinity, we are
  generally trying to find a functions end
  behavior
• When we examine a functions end behavior, we
  want to know if it has horizontal or slant
  asymptotes.
• Recall the following rules for a function of the
  form

3
Limits at Infinity

• If you examine the degree of the numerator and
  denominator, theres one of three possibilities
• n gt m The degree of the numerator is bigger
  than the denominator (top heavy)
• L
• n lt m The degree of the numerator is smaller
  than the denominator (bottom heavy)
• L 0
• n m The degree of the numerator is equal to
  the denominator
• L
• n m 1 The degree of the numerator is
  exactly one more than the the denominator
• L the quotient using long division

There is no horizontal asymptote
The horizontal asymptote is y 0
The horizontal asymptote is y
Not horizontal, but a slant asymptote
4
Limits at Infinity

• Do you remember the limits at infinity for our
  trigonometric functions?

The horizontal asymptote of these functions is y
0

• Remember not to confuse it with the limit as x
  approaches zero!

5
Limits at Infinity

• Example 1 Find all vertical, horizontal, and
  slant asymptotes

• What is the vertical asymptote?

There is a vertical asymptote at x 0

• What is the horizontal asymptote?

There is a horizontal asymptote at y 0

• What is the slant asymptote?

There are no slant asymptotes
6
Limits at Infinity

• Example 2 Find all vertical, horizontal, and
  slant asymptotes

• What is the vertical asymptote?

There are no vertical asymptotes

• What is the horizontal asymptote?

There is a horizontal asymptote at y 0

• What is the slant asymptote?

There are no slant asymptotes
7
Limits at Infinity

• Example 3 Find all vertical, horizontal, and
  slant asymptotes

• What is the vertical asymptote?

There is a vertical asymptote at x 0

• What is the horizontal asymptote?

There is a horizontal asymptote at y 2

• What is the slant asymptote?

There are no slant asymptotes
8
Limits at Infinity

• Example 4 Find all vertical, horizontal, and
  slant asymptotes

• What are the vertical asymptote?

There are vertical asymptotes at x -½ and -1

• What is the horizontal asymptote?

There is a horizontal asymptote at y 2

• What is the slant asymptote?

There are no slant asymptotes
9
Limits at Infinity

• Example 5 Find all vertical, horizontal, and
  slant asymptotes

• What are the vertical asymptote?

There are vertical asymptotes at x -2 and 2

• What is the horizontal asymptote?

There is a horizontal asymptote at y 3

• What is the slant asymptote?

There are no slant asymptotes
10
Limits at Infinity

• Example 6 Find all vertical, horizontal, and
  slant asymptotes

• What are the vertical asymptote?

There is a vertical asymptotes at x -3/2

• What is the horizontal asymptote?

There is a horizontal asymptote at y ½

• What is the slant asymptote?

There are no slant asymptotes
11
Limits at Infinity

• Example 7 Find all vertical, horizontal, and
  slant asymptotes

• What are the vertical asymptote?

There is a vertical asymptotes at x 4

• What is the horizontal asymptote?

There are no horizontal asymptotes

• What is the slant asymptote?

There is a slant asymptote at y 3x 1 by long
division
12
Limits at Infinity

• Example 8 Find all vertical, horizontal, and
  slant asymptotes

• What are the vertical asymptote?

There are no vertical asymptotes

• What is the horizontal asymptote?

There are no horizontal asymptotes

• What is the slant asymptote?

There are no slant asymptotes
13
Limits at Infinity

• Example 9 Heres a tough one Evaluate

Divide everything by x
Apply the limit
14
Homework

• Limits at Infinity
• Read Sections 3.5
• P. 205 1 81 (odd)