3.5 Limits at InfinityObjective: Determine (finite) limits at infinity,

horizontal asymptotes, and infinite limits at infinity.

Miss BattagliaAP Calculus AB/BC

Limits at Infinity

x -∞ -100 -10 -1 0 1 10 100 ∞

f(x) 3 2.9997 2.97 1.5 0 1.5 2.97 2.9997 3

f(x) approaches 3

x decreases without bound x increases without bound

f(x) approaches 3

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 Limits at Infinity

The line y=L is a horizontal asymptote of the graph of f if

or

Definition of a Horizontal Asymptote

If r is a positive rational number and c is any real number, then

Furthermore, if xr is defined when x<0, then

Thm 3.10 Limits at Infinity

Find the limit:

Finding a Limit at Infinity

Find the limit:

Finding a Limit at Infinity

Find each limit.

a. b. c.

A Comparison of Three Rational Functions

1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.

2. 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.

3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

Guidelines for Finding Limits at +∞ of Rational Functions

Find each limit.

a. b.

A Function with Two Horizontal Asymptotes

Find each limit.

a. b.

Limits Involving Trig Functions

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?

Oxygen Level in a Pond

(1,.5) (2,.6) (10,.9)

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.

Definition of Infinite Limits at Infinity

Find each limit.

a. b.

Finding Infinite Limits at Infinity

Find each limit.

a. b.

Finding Infinite Limits at Infinity

AB/BC Calculus: Read 3.5 Page 205 #1-6, 19-35 odd, Quiz tomorrow

Classwork/Homework