13.4 Limits of Infinite Sequences

13.4 Limits of Infinite Sequences • Sequences with a numerical limit • Sequences with no limit • Sequences with a limit of infinity

Limits • Limits are used to determine how a function, sequence, or series will behave as the independent variable approaches a certain value, often infinity. • Limits are written in the form below. • It is read “The limit of 1 over n as n approaches infinity”.

TABLE What number is each term getting closer to???

13.4 Limits of Infinite Sequences GRAPH

13.4 Limits of Infinite Sequences As n gets larger, gets smaller and smaller. In fact, approaches 0. Therefore, . It follows that

13.4 Limits of Infinite Sequences

Evaluating Limits Algebraically • The example below is read “The limit of 1 over n as n approaches infinity”. • 1st choice of evaluating: simple direct substitution • Example:

Direct substitution does not always work, and often is not allowed because the function is not defined for that particular x. For example, • 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.

Rational Function Reminders where f and g are polynomial functions. The degree of a polynomial is equal to the largest exponent in the equation when written in standard form.

Algebraic Manipulation of Limits (Rational Functions) Method 1: Works only if denominator is a single term. 1) If the denominator is a single term, split the fraction into separate fractions. 2) Reduce. 3) Take Limit. =

Method 2: This works for all infinite limits. 1) Divide each part of the fraction by the highest power of n shown in the denominator. 2) Reduce. 3) Take limit. Some terms will drop out.

Practice Use the fact that to evaluate the following with algebraic methods. Check by graphing.

13.4 Limits of Infinite Sequences Infinity is a concept

Practice Evaluate the following. 0 0 1

Shortcut: Guidelines for Finding Limits at ∞ of Rational Functions less than • If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function is ___. • If the degree of the numerator is _______ the degree of the denominator, then the limit of the rational function is the __________________ _______________________. • If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function _______________. 0 equal to ratio of the leading coefficients greater than is infinite

Evaluate by looking, then show you are correct with algebra. 0 2 1 3 ∞

Homework Page 496 #1-29 odds

13.4 Limits of Infinite Sequences