AP Calculus ABChapter 3, Section 5Limits at Infinitiy2013 - 2014

Limits at Infinity

• Limits that approach infinity discuss “end behavior” meaning what does the function do as x grows or decreases without bound. • Graph the function and determine the end

behavior using limits.

Horizontal Asymptotes• If the end behavior of a function approaches a number, that

number is the horizontal asymptote.

• Definition:The line is a horizontal asymptote of the graph of f if

Using this definition, a function can have at most 2 horizontal asymptotes; one from each direction.

Horizontal Asymptote Tricks• If the higher degree is on top, there is no horizontal

asymptote.• If the higher degree is on bottom, the horizontal asymptote is

y = 0.• If the degree of top and bottom is the same, the horizontal

asymptote is the ratio of the leading coefficients.

Limit properties• Limits at infinity have many of the same properties at limits

from chapter 1. For example:

and

Similar properties hold for limits at .

Finding a Limit at Infinity• Find the limit:

Finding a Limit at Infinity• Find the limit:

Finding a Limit at Infinity• Find the limit:

Finding a Limit at Infinity• Find the limit:

Finding a Limit at Infinity• Find the limit:

Functions with 2 Horizontal Asymptotes

• Find the limit:

Functions with 2 Horizontal Asymptotes

• Find the limit:

Limits Involving Trigonometric Fuctions

• Find each limit:

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?

Ch. 3.5 Homework• Pg. 205 – 207, #’s: 3, 5, 7, 15, 17, 21, 27, 33, 59, 89

• 10 total problems