3-5: Limits at Infinity Objectives: ©2003 Roy L. Gover Discuss the end behavior of functions on an infinite interval. Understand horizontal asymptotes and their relation to limits at infinity

Example Find the limit if it exists: In chapter 1, we had this problem:

Example Find the limit if it exists: How does this problem differ from the previous problem?

Important Idea The above symbols describe the increasing or decreasing of a value without bound. Infinity, however, is not a value.

Example Use the table feature of your calculator to estimate: if it exists.

All rules concerning limits still apply. See page 71 of your text: Important Idea

Definition means for any real number >0 (no matter how small) there exists a real number M >0 such that whenever x > M

Definition The absolute value of the difference between f(x) and L

Important Idea means f(x) can be arbitrarily close (as close as you like) to L by choosing a sufficiently large x.

Analysis x f(x) L M For x > M, f(x) is within of L

Definition If it exists, means y = L is a horizontal asymptote L

Important Idea Theorem 3.10: where c is any real number and r is a positive rational number

Example Find the limit, if it exists:

Example Find the limit, if it exists:

Example Find any horizontal asymptotes for:

Example Find the limit, if it exists: Indeter- minate form Divide top & bottom by highest power of x in denominator.

Try This Find the limit, if it exists: DNE or

Try This Find the limit, if it exists: 0

Try This Find the limit, if it exists:

Analysis In the last 3 examples, do you see a pattern? The highest power term is most influential.

Try This Find the limit if it exists:

Example Functions may approach different asymptotes as and as Consider each limit separately…

Example As eventually x >0. Divide radical by and divide non-radical by x. Find the limit:

Example As eventually x <0. Divide radical by and divide non-radical by x. Find the limit:

Try This Using algebraic techniques, find the limit if it exists. Confirm your answer with your calculator.

Solution

Analysis The sine function oscillates between +1 and -1 1

Analysis 1 The limit does not exist due to oscillation.

Analysis Consider

Analysis and Therefore, by the squeeze theorem,

Lesson Close After this lesson, you should be able to evaluate limits at infinity both algebraically and graphically.

Assignment 203/1-21 odd & 29