If we plugged in  directly, we would get an indeterminate value of .

We can use L'Hopital's rule to fix this. We take the derivate of the top and bottom and reevaluate the same limit.

.

We still can't evaluate the limit of the new expression, so we do it one more time.

Subbing in zero into  will give you , so we can try to use L'hopital's Rule to solve.

First, let's find the derivative of the numerator. 

 is in the form , which has the derivative , so its derivative is . 

 is in the form , which has the derivative , so its derivative is .

The derivative of  is  so the derivative of the numerator is .

In the denominator, the derivative of  is , and the derivative of  is . Thus, the entire denominator's derivative is .

Now we take the 

, which gives us . 

When you try to solve the limit using normal methods, you find that the limit approaches zero in the numerator and denominator, resulting in an indeterminate form "0/0". 

In order to evaluate the limit, we must use L'Hopital's Rule, which states that:

when an indeterminate form occurs when evaluting the limit.

Next, simply find f'(x) and g'(x) for this limit:

The derivatives were found using the following rules:

, 

Next, using L'Hopital's Rule, evaluate the limit using f'(x) and g'(x):

Through direct substitution, we see that the limit becomes

which is in indeterminate form.

As such we can use l'Hospital's Rule, which states that if the limit 

is in indeterminate form, then the limit is equivalent to

Taking the derivatives we use the power rule which states 

Using the power rule the limit becomes 

As such the limit exists and is

Through direct substitution, we see that the limit becomes

which is in indeterminate form.

As such we can use l'Hospital's Rule, which states that if the limit 

is in indeterminate form, then the limit is equivalent to

Taking the derivatives we use the trigonometric rule which states 

where  is a constant.

Using l'Hospital's Rule we obtain

And through direct substitution we find

As such the limit exists and is

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:

, , 

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:

,