To calculate the limit we first plug the limit value into the numerator and denominator of the expression. When we do this we get , which is undefined. We now use L'Hopital's rule which says that if  and  are differentiable and

,

then 

.

In this case we are calculating 

so

and

.

We calculate the derivatives and find that 

and

.

Thus

.

If we plugged in the limit value, , directly we would get the indeterminate value . We now use L'Hopital's rule which says that if  and  are differentiable and

,

then 

.

The limit we wish to evaluate is 

,

so we have

and

.

We differentiate both functions and find

and

.

Now using L'Hopital's rule we find 

. 

When we plug the limit value in to this expression we still get the indeterminate value . Thus we must use L'Hopital's rule again.

and

.

Using L'Hopital's rule again we find

.

To calculate the limit we first plug the limit value into the numerator and denominator of the expression. When we do this we get , which is undefined. We now use L'Hopital's rule which says that if  and  are differentiable and

,

then 

.

We are evaluating the limit

so we have

and

.

We differentiate these functions and find

and

.

Using L'Hopital's rule we see

.

To calculate the limit we first plug the limit value into the numerator and denominator of the expression. When we do this we get , which is undefined. We now use L'Hopital's rule which says that if  and  are differentiable and

,

then 

.

We are evaluating the limit

.

In this case we have

and

.

We differentiate both functions and get

and

.

Using L'Hopital's rule we find

.

To evaluate the limit, we must compare the numerator and denominator. The numerator and denominator both have a term containing a third degree term. However, dividing their coefficients is not enough to determine the limit. There is an exponential term in the numerator, which grows faster than any polynomial terms. It dominates the function, and the limit goes to .

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

Using the rule for our limit, we get

We used the following rule to find the derivative:

When evaluating the limit using normal methods, we receive the indeterminate form . When this occurs, we must use L'Hopital's Rule to solve the limit. The rule states that

.

Using the rule, we get

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

.

Using the rule for our limit, we get

We used the following rules to find the derivatives:

, 

Evaluating the limit to begin with gets us , which is undefined. We can solve this problem using L'Hospital's rule. Taking the derivative of the numerator and denominator with respect to n, we get . The limit is still undefined. Another application of the rule gets us , which evaluated at  is in fact .

In evaluating the limit to begin with, you get , which is undefined. Applying L'Hospitals Rule, we take the derivative of both the numerator and denominator with respect to n. The first derivative gets us , which is still improper. Another application of the rule will get us , the correct solution.