If we try to directly plug in the limit value into the function, we get

Because the limit is of the form , we can apply L'Hopital's rule to "simplify" the limit to

.

Now if we directly plug in 0 again, we get

.

In order to determine the limit, substitute  to determine whether the expression is indeterminate.  

Use the L'Hopital's rule to simplify.  Take the derivative of the numerator and denominator separately, and reapply the limit.  

Substitute 

By substitution, the limit will yield an indeterminate form .  L'Hopital can be used in this scenario.

Take the derivative of the numerator and denominator separately, and then reapply the limit.

The answer is .

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):

Rewrite the expression.

By substitutition, we will get the indeterminate form .

The L'Hopital's rule can be used to solve for the limit.  Write the L'Hopital's rule.

Apply this rule twice.

When evaluating the limit using normal methods (substitution), you find that the indeterminate form of  is reached. To evaluate the limit, we can use L'Hopital's Rule, which states that:

So, we find the derivative of the numerator and denominator, which is

 and , respectively.

The derivatives were found using the following rules:

, 

When we evaluate this new limit, we find that

.

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