3 5 Limits at Infinity Objective Determine finite

3. 5 Limits at Infinity Objective: Determine (finite) limits at infinity, horizontal asymptotes, and infinite limits at infinity. Miss Battaglia AP Calculus AB/BC

Limits at Infinity x decreases without bound x increases without bound x -∞ 0 f(x) 3 -100 -1 1 10 100 2. 9997 2. 97 1. 5 0 1. 5 2. 97 2. 9997 f(x) approaches 3 ∞ 3 f(x) approaches 3

Definition of Limits at Infinity Let L be a real number. 1. The statement means that for each ε>0 there exists an M>0 such that |f(x)-L|<ε whenever x>M. 2. The statement means that for each ε>0 there exists an N<0 such that |f(x)-L|<ε whenever x<N.

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

Thm 3. 10 Limits at Infinity If r is a positive rational number and c is any real number, then Furthermore, if xr is defined when x<0, then

Finding a Limit at Infinity Find the limit:

A Comparison of Three Rational Functions Find each limit. a. b. c.

Guidelines for Finding Limits at +∞ of Rational Functions 1. 2. 3. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

A Function with Two Horizontal Asymptotes Find each limit. a. b.

Limits Involving Trig Functions Find each limit. a. b.

Oxygen Level in a Pond Suppose that f(t) measures the level of oxygen in a pond, where f(t)=1 is the normal (unpolluted) level and the time t is measured in weeks. When t=0, organic waste is dumped into the pond, and as the waste material oxidizes, the level of oxygen in the pond is What percent of the normal level of oxygen exists in the pond after 1 week? After 2 weeks? After 10 weeks? What is the limit as t approaches infinity? (1, . 5) (2, . 6) (10, . 9)

Definition of Infinite Limits at Infinity Let f be a function defined on the interval (a, ∞) 1. The statement means that for each positive number M, there is a corresponding number N>0 such that f(x)>M whenever x>N. 2. The statement means that for each negative number M, there is a corresponding number N>0 such that f(x)<M whenever x>N.

Finding Infinite Limits at Infinity Find each limit. a. b.

Classwork/Homework �AB/BC Calculus: Read 3. 5 Page 205 #1 -6, 19 -35 odd, Quiz tomorrow