Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
The Limit is Infinite
The function is undefined at
, but we can still ask about the behavior of
for values of
"close to"
. Fig. 6 indicates that if
is very small, close to
, then
is very large. As the values of
get closer to
, the values of
grow larger and can be made as large as we want by picking
to be close enough to
. Even though the values of
are not approaching any number, we use the "infinity" notation to indicate that the values of
are growing without bound, and write
.
Fig. 6
The values of do not equal "infinity:"
means that the values of
can be made arbitrarily large by picking values of
very close to
.
The limit, as , of
is slightly more complicated. If
is close to
, then the value of
can be a large positive number or a large negative number, depending on the sign of
.
The function does not have a (two-sided) limit as
approaches
, but we can still ask about one-sided limits:
Solution: (a) As , then
and
. Since the denominator is approaching
we cannot use the Main Limit Theorem, and we need to examine the functions more carefully. If
, then
so
. If
is close to
and slightly larger than
, then the ratio of
to
is the ratio
. As
gets closer to
is
. By taking
closer to
, the denominator gets closer to
but is always positive, so the ratio gets arbitrarily large and negative:
.
(b) As , then
and
gets arbitrarily close to
, and
is negative. The value of the ratio
is
:
.