1. 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.

2. 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.

3. Horizontal Asymptote

4. 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.

5. Finding a Limit as x Approaches Infinity
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.

7. Using Theorem 5
Find

9. 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.

10. 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.

11. 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.

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

14. 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,

15. Using Substitution
Find