# 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

.

Evaluate: