We first considered exponential functions because of their rapid growth. Now we would like to say
more precisely how fast they grow by comparing them to some other
functions. In particular, we will show that the function
grows much more rapidly than any power function
where
is a positive
exponent.
First, we'll create a measurement of the relative rates of growth by
considering the ratio of to . That means we will consider the function
Notice that when this function is small, it means that the
denominator, , is larger than the numerator, . We will show that we can make this ratio as small as
we like by increasing the values of
that we
consider. This will be our first principle: to study what happens to
this ratio
as becomes very large.
Statement of Results
As mentioned above, we will show you that the exponential function
grows much faster than the power function as becomes
very large. We will write this as
which is another of expressing the fact that the ratio between the
two functions can be made as small as we like by considering large
values of . What is remarkable is that this is true
for any power function. We saw early in the course that the graphs of
power functions can become very steep
when the exponent is large. Our result here is saying that an
exponential function will always eventually be much larger than that
power function.
In addition, it is true that logarithms grow much more slowly
than power functions. That is to say
for any power function. We will not explicitly show this to you but
you may wish to think about it as following naturally from the fact
that the natural logarithm function is the inverse of the exponential
function.
To accomplish our aim, we are going to use some of the techniques
we have already developed. However, you may find that this is a
little more difficult than what we have done up to this point. If so,
that's all rightyou may view this section as optional. Others may
gain something from seeing a slightly more
sophisticated application of our ideas.

The derivative of
Comparison with an exponential
