# Calculus 2 : L'Hospital's Rule

## Example Questions

### Example Question #61 : L'hospital's Rule

Explanation:

The first step to computing a limit is direct substitution:

Now, we see that this is in the form of L'Hopital's Rule.  For those problems, we take a derivative of both the numerator and denominator separately.  Remember,  is simply a constant!

Now, we can go back and plug in the original limit value:

### Example Question #62 : L'hospital's Rule

Explanation:

The first step to computing a limit is direct substitution:

Now, we see that this is in the form of L'Hopital's Rule.  For those problems, we take a derivative of both the numerator and denominator separately.  Remember,  is simply a constant!

Now, we can go back and plug in the original limit value:

.

However, we still have to rationalize the denominator.  Therefore:

.

### Example Question #63 : L'hospital's Rule

Evaluate the following limit by L'hospital's Rule

Undefined

Explanation:

Recall L'hospital's Rule for an indeterminate limit is as follows:

Since  is an indeterminate limit, one must use L'hospital's  rule.

Therefore the question now becomes,

### Example Question #64 : L'hospital's Rule

Solve the limit:

Explanation:

Notice if we try to plug infinity into the limit we get  so we apply L'Hospital's rule.

We then take the derivative of the top and the bottom and get

### Example Question #65 : L'hospital's Rule

Evaluate

Explanation:

Substituting 0 directly into x gives

,

which is an indeterminant form that allows the use of L'hospital's rule. Applying L'hospital's rule, we get

Using the the pythagorean trig. identity, , we rewrite the limit as

Now we plug 0 in for x.

### Example Question #66 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of , it returns the indeterminate form . We start by rewriting the expression

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. First we will rewrite the expression.

Where,

So

### Example Question #67 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of x = -1, 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 #68 : 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 #69 : L'hospital's Rule

Evaluate the limit:

Explanation:

If we evaluate the expression with the limit of x = 8, 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 #70 : 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, so we simply go through L'Hospital's Rule a second time.