Calculus 2 : New Concepts

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #52 : L'hospital's Rule

Evaluate:

Possible Answers:

The limit does not exist

Correct answer:

Explanation:

When evaluating the limit using normal methods, we receive the indeterminate form . When this occurs, we must use L'Hopital's Rule to solve the limit. The rule states that

.

Using the rule, we get

Example Question #53 : L'hospital's Rule

Evaluate the limit:

Possible Answers:

The limit does not exist

Correct answer:

Explanation:

When evaluating the limit using normal methods, we get the indeterminate form . When this occurs, we must use L'Hopital's Rule to evaluate the limit. The rule states that

.

Using the rule for our limit, we get

We used the following rules to find the derivatives:

Example Question #54 : L'hospital's Rule

Evaluate 

Possible Answers:

Correct answer:

Explanation:

Evaluating the limit to begin with gets us , which is undefined. We can solve this problem using L'Hospital's rule. Taking the derivative of the numerator and denominator with respect to n, we get . The limit is still undefined. Another application of the rule gets us , which evaluated at  is in fact .

Example Question #55 : L'hospital's Rule

Evaluate 

Possible Answers:

Correct answer:

Explanation:

In evaluating the limit to begin with, you get , which is undefined. Applying L'Hospitals Rule, we take the derivative of both the numerator and denominator with respect to n. The first derivative gets us , which is still improper. Another application of the rule will get us , the correct solution. 

Example Question #56 : L'hospital's Rule

Evaluate 

Possible Answers:

Correct answer:

Explanation:

Evaluating the limit to begin with, we get , which is undefined. Using L'Hospital's rule to solve, we take the derivative of the numerator and denominator of the expression. In doing so, we get . Evaluating the new limit, we still get . Another application of L'Hospitals rule gives us . We can now solve the limit, which is 

Example Question #57 : L'hospital's Rule

Find the limit

Possible Answers:

Correct answer:

Explanation:

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Example Question #58 : L'hospital's Rule

Use L'Hospital's rule to evaluate

.

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

To use L'hospital's rule, evaluate the limit of the numerator of the fraction and the denominator separately. If the result is , or , take the derivative of the numerator and the denominator separately, and try to evaluate the limit again.

 

 (L'hospital's rule) 

 (L'hospital's rule again) 

 (L'hospital's rule again) 

 

Example Question #59 : L'hospital's Rule

Use L'Hospital's Rule to evaluate

 

Possible Answers:

The limit does not exist.

Correct answer:

Explanation:

To use L'hospital's rule, evaluate the limit of the numerator of the fraction and the denominator separately. If the result is , or , take the derivative of the numerator and the denominator separately, and try to evaluate the limit again.

  

 

 (L'hospital's Rule) 

 (L'hospital's Rule. Here the derivative of the numerator involves the Product Rule)

 

Example Question #60 : L'hospital's Rule

Use L'hospital's Rule to evaluate

.

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

To use L'hospital's rule, evaluate the limit of the numerator of the fraction and the denominator separately. If the result is , or , take the derivative of the numerator and the denominator separately, and try to evaluate the limit again.

 

 

 (L'hospital's Rule)

.

So the answer is .

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:

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