# Calculus 2 : L'Hospital's Rule

## Example Questions

### Example Question #41 : L'hospital's Rule

Evaluate the following limit, if possible:

.

The limit does not exist

Explanation:

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

.

### Example Question #42 : L'hospital's Rule

Evaluate the following limit, if possible:

The limit does not exist

The limit does not exist

Explanation:

The limit does not exist. When we first plug in the limit value, 0, we get 0/0 and indeterminate form. We may try to use L'Hopital's rule. However, we quickly discover that

is not differentiable at x=0, so we cannot use L'Hopital's rule. We then consider the limits from the left and right. We determine that

and

.

Thus the two sided limit does not exist.

### Example Question #43 : L'hospital's Rule

Evaluate the following limit, if it exists:

Explanation:

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 trying to evaluate the limit

,

so we have

and

.

We differentiate both of these functions and find

and

.

Using L'Hopital's rule we see

.

Now when we plug the limit value into the expression we get , so

.

### Example Question #44 : L'hospital's Rule

Evaluate the following limit, if possible:

.

The limit does not exist.

Explanation:

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 evaluate the derivatives of these two functions and find

and

.

Thus, using L'Hopital's we find

.

However, when we plug the limit value into this second expression we still end up with the indeterminate value . So we use L'Hopital's again.

and

.

Using L'Hopital's rule again we see

.

Plugging in the limit value now we see that the limit evaluates to

### Example Question #45 : L'hospital's Rule

Evaluate the following limit, if possible:

The limit does not exist

Explanation:

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 trying to evaluate the limit

so we have

and

.

We calculate the derivatives of these functions and get

and

.

Using L'Hopital's rule we find

.

Now when we plug in the limit value we get , so the limit evaluates to 0.

### Example Question #46 : L'hospital's Rule

Evaluate the following limit, if possible:

.

The limit does not exist

Explanation:

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

.

### Example Question #47 : L'hospital's Rule

Evaluate the following limit, if possible:

The limit does not exist

Explanation:

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

.

### Example Question #41 : L'hospital's Rule

Evaluate the following limit, if possible:

The limit does not exist

Explanation:

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

.

### Example Question #49 : L'hospital's Rule

Evaluate the following limit, if possible:

.

The limit does not exist

Explanation:

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

.

### Example Question #50 : L'hospital's Rule

Evaluate:

Explanation:

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 .