limits and continuity

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