p192 Section 3.5: Limits at Infinity
Definition of Limits at Infinity: Let L be a real number
The limit of a function as x approaches infinity (or negative infinity) is the number, L, that the function approaches.
Definition of Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if or
** The graph of a function can have at most two horizontal asymptotes ­ one to the right and one to the left.
** Limits at infinity have the same properties of limits discussed in Ch 2
Theorem 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
1
Example 1: Evaluating a Limit at Infinity
Find the limit:
Example 2: Evaluating a Limit at Infinity
2
Example 3: A Comparison of Three Rational Functions
Find each of the limits
a
b
c
3
Guidelines for finding Limits at Infinity of 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 (HA y = 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. (HA y = a/b)
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 (No HA) Example 4: A function with two horizontal asymptotes
a
b
4
Example 5: Limits involving Trigonometric Functions
a
b
Example 6: 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?
5
Definition of Infinite Limits at Infinity: Let f be a function defined on the interval (a, ∞)
means that as x increases without
1. The statement bound, f(x) also increases without bound ­ there is no finite limit at infinity.
2. The statement
bound, f(x) decreases without bound.
means that as x increases without ** No polynomial function has a finite limit at infinity.
Example 7: Finding Infinite Limits at Infinity
a
b
6
Example 8: Finding Infinite Limits at Infinity
a
b
#13
Find a
b
c
7
#17 a.
b
c
#27
8
#29
#31
#33. Use a graphing utility to graph the function and verify that it has two horizontal asymptotes.
9