Sec 4: Limits at Infinity

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Presentation transcript:

Sec 4: Limits at Infinity

Definition of a Horizontal Asymptote The line y = L is a horizontal asymptote of the graph of f if or

Ex 1: Find the limits numerically and then state the horizontal asymptote(s) -∞ -100 -10 -1 1 10 100 ∞ f(x) ? Horizontal Asymptote(s):

Theorem: Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if is defined when x < 0, then

Ex 2: Evaluate a simple Limit at Infinity

Ex 3: Evaluate

Shortcuts for finding Limits at Infinity If the degree of the numerator is equal to the degree of the denominator, then the limit is the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, then the limit is 0. If the degree of the numerator is more than the degree of the denominator, then the limit is ∞.

Ex 3: Evaluate using the shortcuts for limits at infinity B. C.

HOMEWORK Pg 199 #13 - 24

Ex 4: Limits of Non-Rational Functions With Non-Rational functions, sometimes there will be 2 horizontal asymptotes, which means two different limits. A. B. *for -∞, change bottom fraction to

Ex 5: Limits at Infinity using Reasoning A. C. B. D.

HOMEWORK Pg 199 #7-12 (do algebraically)