Before proceeding, consider the graph of
shown in
[link] . As
and
the graph of
appears almost linear. Although
is certainly not a linear function, we now investigate why the graph of
seems to be approaching a linear function. First, using long division of polynomials, we can write
Since
as
we conclude that
Therefore, the graph of
approaches the line
as
This line is known as an
oblique asymptote for
(
[link] ).
The graph of the rational function
approaches the oblique asymptote
We can summarize the results of
[link] to make the following conclusion regarding end behavior for rational functions. Consider a rational function
where
If the degree of the numerator is the same as the degree of the denominator
then
has a horizontal asymptote of
as
If the degree of the numerator is less than the degree of the denominator
then
has a horizontal asymptote of
as
If the degree of the numerator is greater than the degree of the denominator
then
does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as
where the degree of
is less than the degree of
As a result,
Therefore, the values of
approach zero as
If the degree of
is exactly one more than the degree of
the function
is a linear function. In this case, we call
an oblique asymptote.
Now let’s consider the end behavior for functions involving a radical.
Determining end behavior for a function involving a radical
Find the limits as
and
for
and describe the end behavior of
Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of
To determine the appropriate power of
consider the expression
in the denominator. Since
for large values of
in effect
appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by
Then, using the fact that
for
for
and
for all
we calculate the limits as follows:
Therefore,
approaches the horizontal asymptote
as
and the horizontal asymptote
as
as shown in the following graph.
This function has two horizontal asymptotes and it crosses one of the asymptotes.
Determining end behavior for transcendental functions
The six basic trigonometric functions are periodic and do not approach a finite limit as
For example,
oscillates between
(
[link] ). The tangent function
has an infinite number of vertical asymptotes as
therefore, it does not approach a finite limit nor does it approach
as
as shown in
[link] .
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