Find the equation for each horizontal asymptote. 1. 2. 3.

12 Limits and an Introduction to Calculus Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 12.4 LIMITS AT INFINITY AND LIMITS OF SEQUENCES Copyright © Cengage Learning. All rights reserved.

What You Should Learn 12-4 Evaluate limits of functions at infinity. Find limits of sequences. We will do this later.

Limits at Infinity and Horizontal Asymptotes

Limits at Infinity and Horizontal Asymptotes There are two basic problems in calculus: finding tangent lines and finding the area of a region. We have seen how limits can be used to solve the tangent line problem. In this section and the next, you will see how a different type of limit, a limit at infinity, can be used to solve the area problem. To get an idea of what is meant by a limit at infinity, consider the function given by

Limits at Infinity and Horizontal Asymptotes The graph of f is shown in Figure 12.30. Figure 12.30

Limits at Infinity and Horizontal Asymptotes From earlier work, you know that is a horizontal asymptote of the graph of this function. Using limit notation, this can be written as follows. These limits mean that the value of f (x) gets arbitrarily close to as x decreases or increases without bound. Horizontal asymptote to the left Horizontal asymptote to the right

Limits at Infinity and Horizontal Asymptotes

Limits at Infinity and Horizontal Asymptotes To help evaluate limits at infinity, you can use the following definition.

Example 1 – Evaluating a Limit at Infinity Find the limit. Solution: Use the properties of limit

Example 1 – Solution = 4 – 3(0) = 4 cont’d = 4 – 3(0) = 4 So, the limit of f (x) = 4 – as x approaches is 4.

You Do Exercise 9, page 887

A Mr. Green time-saver:

Work through how to do the graph by hand. You Do Exercise 19, page 887 Work through how to do the graph by hand.

Graph of

Limits at Infinity and Horizontal Asymptotes

Anything with a bar over it means the average.

This is so old! Let’s say it’s a smart phone! You Do Exercise 55, page 888

You Do Exercise 55, page 888 a) c) As more PDAs are produced, the average cost per unit will approach $13.50. b)

Exit Why is it that when we evaluate limits toward infinity of rational functions, that we can find the limit simply by using the terms of highest degree in the numerator and the denominator?