# Calculus 2 : L'Hospital's Rule

## Example Questions

### Example Question #71 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

Therefore,

If we rewrite the limit with L'Hospital's Rule,

### Example Question #72 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of x = -3, it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

Therefore,

If we rewrite the limit with L'Hospital's Rule,

### Example Question #73 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

So,

If we rewrite the limit with L'Hospital's Rule,

This is another indeterminate form, so we simply go through L'Hospital's Rule a second time.

### Example Question #74 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

Therefore,

If we rewrite the limit with L'Hospital's Rule,

This is another indeterminate form, and since , we can multiple the fraction by

### Example Question #75 : L'hospital's Rule

Use L'Hospital's rule to find  .

Explanation:

L'Hospital's rule state that if , or  , then

To solve this problem, we must first see if L'Hospital's rule applies, by substitution.

Since, we can use L'Hospital's rule.  Take the derivative of the top and bottom of the fraction, gives us

This means we can use L'Hospital's rule again.  Taking the derivative of the top and bottom of the fraction a second time gives us

### Example Question #76 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

Therefore,

If we rewrite the limit with L'Hospital's Rule,

### Example Question #77 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

So,

If we rewrite the limit with L'Hospital's Rule,

### Example Question #78 : L'hospital's Rule

Use L'Hospital's rule to find  .

Explanation:

L'Hospital's rule state that if , or  , then

To solve this problem, we must first see if L'Hospital's rule applies, by substitution.

Since, we can use L'Hospital's rule.  Take the derivative of the top and bottom of the fraction, gives us

### Example Question #79 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form .

This also returns an indeterminate form of .

We can instead use L’Hospital’s Rule to evaluate, using the form:

Where,

So,

If we rewrite the limit with L'Hospital's Rule,

Then we can conclude

### Example Question #80 : L'hospital's Rule

Use L'Hospital's rule to find  .