1. 2.2 Infinite Limits and Limits at Infinity

2. Infinite limits Definition: The notation
(read as “the limit of of f(x) , as x approaches a, is infinity”)
means that the values of f(x) can be made arbitrarily large by taking x sufficiently close to a (on either side of a) but not equal to a.
Note: Similar definitions can be given for negative infinity and the one-sided infinite limits.
Example:

3. Vertical Asymptotes
Definition: The line x=a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true:
Example:
x=0 is a vertical asymptote for y=1/x2

4. Limits at Infinity
Definition: Let f be a function defined on some interval (a,). Then
(read as “the limit of of f(x) , as x approaches infinity, is L”)
means that the values of f(x) can be made as close to L as we like by taking x sufficiently large.
Note: Similar definitions can be given for negative infinity.
Example:

5. Horizontal Asymptotes
Definition: The line y=L is called a horizontal asymptote of the curve y=f(x) if either
Example:
y=0 is a horizontal asymptote for y=1/x2
Generally, if n is a positive integer, then

6. Infinite Limits at Infinity
Definition: The notation
is used to indicate that the values of f(x) become large as x becomes large.
Similarly, we can have
Example: