When we evaluate the limit using normal methods (substitution), we get an indeterminate form . To evaluate the limit, we must use L'Hopital's Rule, which states that:

Now we compute the derivative of the numerator and denominator of the original function:

, 

The derivatives were found using the following rules:

, 

Now, evaluate the limit:

When we evaluate the limit using normal methods, we get , an indeterminate form. So, to evaluate the limit, we must use L'Hopital's Rule, which states that when we receive the indeterminate form of the type above (or ):

So, we must find the derivative of the top and bottom functions:

, 

The derivatives were found using the following rules:

, , 

Now, rewrite the limit and evaluate:

When evaluating the limit using normal methods, we get the indeterminate form . When this occurs, or when  occurs, we must use L'Hopital's Rule to evaluate the limit. The rules states 

So, we must find the derivative of the numerator and denominator:

The derivatives were found using the following rules:

, , 

Now, using the above formula, evaluate the limit:

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

When evaluting the limit using normal methods, we find that the indeterminate form  is reached. When this (or ) happens, we use L'Hopital's Rule to evaluate the limit:

So, we must find the derivative of the numerator and denominator:

When we plug these into the formula and evaluate the limit we get:

We being by attempted to plug in  into our given function.

Since this would yield , we can use L'Hospital's Rule to help us find the limit.

Replace the numerator and the denominator of our function with their respective derivatives, and we get

Hence the answer 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  again yields the indeterminate form

So we apply L'Hospital's rule again

Evaluating for  yields

As such

and thus

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