3.5 Limits at Infinity Determine limits at infinity

Determine the horizontal asymptotes, if any, of the graph of function.

Standard 4.5a

Do Now: Complete the table.

x -∞ -100

-10 -1 0 1 10 100 ∞

f(x)

x -∞ -100

-10 -1 0 1 10 100 ∞

f(x) 2 1.99

1.96

.667

0 .667

1.96

1.99

2

x decreases x increases

f(x) approaches 2 f(x) approaches 2

Limit at negative infinity

Limit at positive infinity

We want to investigate what happens when functions go

To Infinity and

Beyond…

Definition of a Horizontal Asymptote

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

Limits at InfinityIf r is a positive rational number and c is any real number, then

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

Finding Limits at Infinity

Finding Limits at Infinity

is an indeterminate form

Divide numerator and denominator by highest degree of x

Simplify

Take limits of numerator and denominator

Guidelines for Finding Limits at

± ∞ of Rational Functions1. If the degree of the numerator is < the

degree of the denominator, then the limit is 0.

2. If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients.

3. If the degree of the numerator is > the degree of the denominator, then the limit does not exist.

For x < 0, you can write

Limits Involving Trig FunctionsAs x approaches ∞, sin x oscillates between -1 and 1. The limit does not exist.

By the Squeeze Theorem

Sketch the graph of the equation using extrema, intercepts, and asymptotes.