Suppose the functions
and
both approach infinity as
Although the values of both functions become arbitrarily large as the values of
become sufficiently large, sometimes one function is growing more quickly than the other. For example,
and
both approach infinity as
However, as shown in the following table, the values of
are growing much faster than the values of
Comparing the growth rates of
And
In fact,
As a result, we say
is growing more rapidly than
as
On the other hand, for
and
although the values of
are always greater than the values of
for
each value of
is roughly three times the corresponding value of
as
as shown in the following table. In fact,
Comparing the growth rates of
And
In this case, we say that
and
are growing at the same rate as
More generally, suppose
and
are two functions that approach infinity as
We say
grows more rapidly than
as
if
On the other hand, if there exists a constant
such that
we say
and
grow at the same rate as
Next we see how to use L’Hôpital’s rule to compare the growth rates of power, exponential, and logarithmic functions.
Comparing the growth rates of
And
For each of the following pairs of functions, use L’Hôpital’s rule to evaluate
Since
and
we can use L’Hôpital’s rule to evaluate
We obtain
Since
and
we can apply L’Hôpital’s rule again. Since
we conclude that
Therefore,
grows more rapidly than
as
(See
[link] and
[link] ).
An exponential function grows at a faster rate than a power function.
Growth rates of a power function and an exponential function.
Since
and
we can use L’Hôpital’s rule to evaluate
We obtain
Thus,
grows more rapidly than
as
(see
[link] and
[link] ).
A power function grows at a faster rate than a logarithmic function.
Growth rates of a power function and a logarithmic function
Using the same ideas as in
[link] a. it is not difficult to show that
grows more rapidly than
for any
In
[link] and
[link] , we compare
with
and
as
The exponential function
grows faster than
for any
(a) A comparison of
with
(b) A comparison of
with
An exponential function grows at a faster rate than any power function
Similarly, it is not difficult to show that
grows more rapidly than
for any
In
[link] and
[link] , we compare
with
and
The function
grows more slowly than
for any
as
A logarithmic function grows at a slower rate than any root function
Key concepts
L’Hôpital’s rule can be used to evaluate the limit of a quotient when the indeterminate form
or
arises.
L’Hôpital’s rule can also be applied to other indeterminate forms if they can be rewritten in terms of a limit involving a quotient that has the indeterminate form
or
The exponential function
grows faster than any power function
The logarithmic function
grows more slowly than any power function
For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
For the following exercises, use a calculator to graph the function and estimate the value of the limit, then use L’Hôpital’s rule to find the limit directly.
