Limits and Continuity Chapter 2 Aim 2 2

Limits and Continuity Chapter 2:

Aim #2. 2 How do we solve limits as x approaches +/- infinity? • We use infinity when the values in its domain or range outgrow all finite bounds. • Graph y = 1/x • What happens as x approaches to + infinity? • What happens as x approaches to - infinity?

Definition: Horizontal Asymptote

Example 1: Looking for Horizontal Asymptotes

Graph the following f(x)

Theorem 5

Example 5: Using the Theorem 5

Infinite Limits as x ---a • Graph the function f(x)= 1/x. What happens as x approaches 0 from the right? • From the left? What does y approach?

Definition: Vertical Asymptotes

Example 4: Finding the Vertical Asymptote What does f(x) approach as x approaches 0 from the right? From the left?

Example 5: Finding Vertical Asymptotes • What are the vertical asymptotes of • f(x)= tan x?

End Behavior Models • For numerically large values of x, we can sometimes model the behavior of a complicated function by a simpler one that acts virtually the same way.

Example 6:

Definition End Behavior Models

Example 7:

Seeing Limits as x approaches +/infinity • We can investigate the graph of y =f(x) as x approaches +/- infinity by investigating the graph of y =f(1/x) as x approaches infinity.

Example 9: Using Substitution

Summary: Answer in complete sentences. • Explain why there is no value L for which lim as x approaches infinity for sin x = L