Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
Vertical Asymptotes
Definition: The vertical line is a vertical asymptote of the graph of
if either or both of the one-sided limits, as
or
, of
is infinite.
If our function is the ratio of a polynomial
and a polynomial
, then the only candidates for vertical asymptotes are the values of
where
. However, the fact that
is not enough to guarantee that the line
is a vertical asymptote of
; we also need to evaluate
. If
and
, then the line
is a vertical asymptote of
. If
and
, then the line
may or may not be a vertical asymptote.
Example 7: Find the vertical asymptotes of and
.
Solution: so the only values which make the denominator
are
and
, and these are the only candidates to be vertical asymptotes.
and
so
and
are both vertical asymptotes of
so the only candidates to be vertical asymptotes are
and