The function
oscillates between
as
The function
does not approach a limit and does not approach
as
Recall that for any base
the function
is an exponential function with domain
and range
If
is increasing over
If
is decreasing over
For the natural exponential function
Therefore,
is increasing on
and the range is
The exponential function
approaches
as
and approaches
as
as shown in
[link] and
[link] .
End behavior of the natural exponential function
The exponential function approaches zero as
and approaches
as
Recall that the natural logarithm function
is the inverse of the natural exponential function
Therefore, the domain of
is
and the range is
The graph of
is the reflection of the graph of
about the line
Therefore,
as
and
as
as shown in
[link] and
[link] .
End behavior of the natural logarithm function
The natural logarithm function approaches
as
Determining end behavior for a transcendental function
Find the limits as
and
for
and describe the end behavior of
To find the limit as
divide the numerator and denominator by
We conclude that
and the graph of
approaches the horizontal asymptote
as
To find the limit as
use the fact that
as
to conclude that
and therefore the graph of approaches the horizontal asymptote
as
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.
Problem-solving strategy: drawing the graph of a function
Given a function
use the following steps to sketch a graph of
Determine the domain of the function.
Locate the
- and
-intercepts.
Evaluate
and
to determine the end behavior. If either of these limits is a finite number
then
is a horizontal asymptote. If either of these limits is
or
determine whether
has an oblique asymptote. If
is a rational function such that
where the degree of the numerator is greater than the degree of the denominator, then
can be written as
where the degree of
is less than the degree of
The values of
approach the values of
as
If
is a linear function, it is known as an
oblique asymptote .
Determine whether
has any vertical asymptotes.
Calculate
Find all critical points and determine the intervals where
is increasing and where
is decreasing. Determine whether
has any local extrema.
Calculate
Determine the intervals where
is concave up and where
is concave down. Use this information to determine whether
has any inflection points. The second derivative can also be used as an alternate means to determine or verify that
has a local extremum at a critical point.
