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 1 10 100 ∞ f(x)

x decreases x increases f(x) approaches 2 f(x) approaches 2 x -∞ -100 1 10 100 ∞ f(x) 2 1.99 1.96 .667 f(x) approaches 2 f(x) approaches 2

Limit at negative infinity Limit at positive infinity

To Infinity and Beyond… 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 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 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 Functions If the degree of the numerator is < the degree of the denominator, then the limit is 0. If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients. 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 Functions As 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.