Calculus 2 : L'Hospital's Rule

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #61 : L'hospital's Rule

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

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

 

Possible Answers:

Undefined

Correct answer:

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: 

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

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.

, giving us the answer.

Example Question #66 : L'hospital's Rule

Evaluate the limit: 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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.

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