We use derivatives to give us a “short-cut” for computing limits.
Derivatives allow us to take problems that were once difficult to solve and
convert them to problems that are easier to solve. Let us consider L’Hôpital’s
rule:
L’Hôpital’s Rule Let and be functions that are differentiable near . If and for all
near , then provided that exists or .
This theorem is somewhat difficult to prove, in part because it incorporates so many
different possibilities, so we will not prove it here.
L’Hôpital’s rule allows us to investigate limits of indeterminate form.
List of Indeterminate Forms
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
-
- This refers to a limit of the form where and as .
In each of these cases, the value of the limit is not immediately obvious. Hence, a
careful analysis is required!
Basic indeterminant forms
Our first example is the computation of a limit that was somewhat difficult
before.
Compute
Set and . Since both and are differentiable functions at , and this
situation is ripe for L’Hôpital’s Rule. Now and L’Hôpital’s rule tells us that
Our next set of examples will run through the remaining indeterminate forms one is
likely to encounter.
Compute
Set and . Both and are differentiable near . Additionally, This
situation is ripe for L’Hôpital’s Rule. Now and L’Hôpital’s rule tells us that
Compute
This doesn’t appear to be suitable for L’Hôpital’s Rule. As approaches
zero, goes to , so the product looks like This product could be anything.
A careful analysis is required. Write Set and . Since both functions are
differentiable near zero and we may apply L’Hôpital’s rule. Write with me and so
One way to interpret this is that since , the function approaches zero much faster
than approaches .
Indeterminate forms involving subtraction
There are two basic cases here, we’ll do an example of each.
Compute
Here we simply need to write each term as a fraction,
Setting and , both functions are differentiable near zero and
We may now apply L’Hôpital’s rule. Write with me and so
Sometimes one must be slightly more clever.
Compute
Again, this doesn’t appear to be suitable for L’Hôpital’s
Rule. A bit of algebraic manipulation will help. Write with me
Now set , . Since both functions are differentiable for large values
of and we may apply L’Hôpital’s rule. Write with me and so
Exponential Indeterminate Forms
There is a standard trick for dealing with the indeterminate forms Given and such
that falls into one of the categories described above, rewrite as and then we rewrite
as Recall that the exponential function is continuous. Therefore Now we
will focus on the limit using L’Hôpital’s rule. We will only give a single
example.
First determine the form of the limit, then compute the limit. Select the correct
choice. The form of the limit is
Write So now look at the limit of the exponent Setting and , both functions are
differentiable for large values of and We may now apply L’Hôpital’s rule. Write and so
Hence,