Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
Practice Answers
Practice 1: As becomes arbitrarily large, the values of
approach
and the values of
approach
.
Practice 3: The completed table is shown in Fig. 12.
Fig. 12
Practice 4: If , then
.
If , then
.
If , then
.
Practice 5 :
(a) .
As the values
, and
so
takes small negative values.
Then the values of are large negative values so we represent the limit as "
".
(b) .
As the values of
, and
so
takes small positive values. As
the values of
.
Then the values of are large positive values so we represent the limit as "
".
(c) .
As , the values of
and
so we need to do more work. The numerator can be factored
and then the rational function can be reduced (since
we know
):
Practice 6 :
(a) .
has vertical asymptotes at
and
.
(b) .
The value is not in the domain of
. If
, then
has a "hole" when
and no vertical asymptotes.
Practice 7: .
has a vertical asymptote at
.
has no horizontal asymptotes.
so
has the linear asymptote
.
Practice 8 :
.
is not defined at
, so
has a vertical asymptote or a "hole" when
.
so
has a "hole" when
.
so
has the nonlinear asymptote
.