1. Warm-up:

2. Section 13-4: Infinite Sequences and Series In this section we will answer…  What makes a sequence infinite?  How can something infinite have a limit?  Is it possible to find the sum of an infinite series?

3. WW hat kind of sequence is it? FF ind the 18 th term. NN ow find the 20 th, 25 th, and 50 th. SS o …the larger n is the more the sequence approaches what? Consider the following sequence: 16, 8, 4, ….

4. Limits  Limits are used to determine how a function, sequence or series will behave as the independent variable approaches a certain value, often infinity.

5. Limits  They are written in the form below:  It is read “The limit of 1 over n as n approaches infinity”.

6. Limits  They are written in the form below:  It is read “The limit of 1 over n as n approaches infinity”.  To evaluate the limit substitute infinity for n:

7. Possible Answers to Infinite Limits  You may get zero or any number.

8. Possible Answers to Infinite Limits You may get infinity. That means no limit exists because it does not approach any single value.  You may get no limit exists because the sequence fluctuates.

9. Possible Answers to Infinite Limits  You may get infinity over infinity.  This is indeterminate; meaning in its present form you can’t tell if it has a limit or not.

10. Possible Answers to Infinite Limits  You may get infinity over infinity.  This is indeterminate; meaning in its present form you can’t tell if it has a limit or not. Let’s do some test values, by using table of values.

11. Algebraic Manipulation of Limits  Method 1: Works only if denominator is a single term. – 1) If denominator is single term, split the into separate fractions. – 2) Reduce – 3) Take Limit

12. Algebraic Manipulation of Limits  Method 2: This works for all infinite limits. – 1) Divide each part of the fraction by the highest power of n shown. – 2) Reduce. – 3) Take limit (Some terms will drop out).

13. Theorem

14. Limits  Use the fact that to evaluate the following:

15. Limits

16. Evaluating limits  Follow these basic steps to evaluate limits.  Plug it in!  Factor, simplify or any other algebraic manipulations.  Check out the table or graph