Derivatives - AP Calculus BC

By substituting the value of , we will find that this will give us the indeterminate form . This means that we can use L'Hopital's rule to solve this problem.

L'Hopital states that we can take the limit of the fraction of the derivative of the numerator over the derivative of the denominator. L'Hopital's rule can be repeated as long as we have an indeterminate form after every substitution.

Take the derivative of the numerator.

Take the derivative of the numerator.

Rewrite the limit and use substitution.

The limit is .

Directly evaluating for yields the indeterminate form

we are able to apply L'Hospital's rule which states that if the limit is in indeterminate form when evaluated, then

As such the limit in the problem becomes

Evaluating for yields

As such

and thus

This important limit from elementary limit theory is usually proven using trigonometric arguments, but it can be shown using L'hopital's rule too.

When evaluating the limit using normal methods, we find that the indeterminate form results. When this occurs, we must use L'Hopital's Rule, which states that for .

Taking the derivative of the top and bottom functions and evaluating the limit, we get

The derivatives were found using the following rules:

, ,

To find dx/dy we must take the derivative of the given function implicitly. Notice the term will require the use of the Product Rule, because it is a composition of two separate functions multiplied by each other. Every other term in the given function can be derived in a straight-forward manner, but this term tends to mess with many students. Remember to use the Product Rule:

Product Rule:

Now if we take the derivative of each component of the given problem statement:

Notice that anytime we take the derivative of a term with y involved we place a "dy/dy" next to it, but this is equal to "1".

So this now becomes:

Now if we place all the terms with a "dx/dy" onto one side and factor out we can solved for it:

This is one of the answer choices.

When evaluating the limit using normal methods (substitution), we get the indeterminate form . When this (or ) occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that when an indeterminate form results from the limit,

Taking the derivatives in our limit, we get

The derivatives were found using the following rules:

, , ,

When evaluating the limit using normal methods, we find that we get the indeterminate form . When this occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that for the limits for which the indeterminate forms result,

Taking the derivatives for our limit, we get

The derivatives were found using the following rules:

,

Simply plugging in into the expression yields the indeterminate form of , so we must resort to using L'Hôpital's rule. We take the derivative of the numerator and denominator, and then look at the limit again.

This time, when we plug in , we do not get an indeterminate form, so we can evaluate the limit by setting :

.

Admittedly a more computationally intensive problem, we begin by trying to plug in into the limit directly, but we get the indeterminate form .

We then apply L'Hôpital's rule and try again:

.

However, plugging in still yields the indeterminate form , so we are forced to apply L'Hôpital's rule a second time:

.

This time, plugging in does not produce an indeterminate form, so we may evaluate the limit by setting as follows:

.