2 2 Limits Involving Infinity Finite Limits as

2. 2 Limits Involving Infinity • Finite Limits as – The symbol for infinity does not represent a real number. – We use infinity to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. • When we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right of the number line. • When we say “the limit of f as x approaches negative infinity”, we mean the limit of f as x moves increasingly far to the left.

• Looking at f(x) = 1/x, we observe: • We can say that the line y = 0 is a horizontal asymptote of the graph of f.

Horizontal Asymptote

Looking for Horizontal Asymptotes • Use graphs and tables to find and identify all horizontal asymptotes of • The horizontal asymptotes are y = 1 and y = -1.

Finding a Limit as x Approaches Infinity • Find • The numerator is decreasing to a very small number. The denominator is increasing to a very large number. This causes the function to approach 0.

Using Theorem 5 • Find

Finding Vertical Asymptotes • Find the vertical asymptotes of f(x) = 1 / x². Describe the behavior to the left and right of each vertical asymptote. • The values of the function approach infinity on either side of x = 0. • The line x = 0 is the only vertical asymptote.

Finding Vertical Asymptotes • The graph of f(x) = tan x = (sin x) / (cos x) has infinitely many vertical asymptotes, one at each point where the cosine is zero.

Modeling Functions for |x| Large • Let and. Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x| large.

Finding End Behavior Models • Find an end behavior model for: a. b.

Finding End Behavior Models • Let f(x) = x + e-x. Show that g(x) = x is a right end behavior model for f while h(x) = e-x is a left end behavior model for f. • On the right, • On the left,

Using Substitution • Find