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.

x

• 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 findand identify all horizontal asymptotes of

• The horizontal asymptotes are y = 1 and y = -1.

lim ( ), lim ( )x xf x f x

2( ) / 1.f x x x

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• Letand . Show that while f and g are quite

different for numerically small values of x, they are virtually identical for |x| large.

4 3 2( ) 3 2 3 5 6f x x x x x 4( ) 3g x x

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 limsin(1/ )x

x