# Calculus 2 : L'Hospital's Rule

## Example Questions

### Example Question #51 : L'hospital's Rule

Evaluate:

Explanation:

When we evaluate the limit using normal methods (substitution), 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 rule to find the derivative:

### Example Question #52 : L'hospital's Rule

Evaluate:

The limit does not exist

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:

The limit does not exist

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

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

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

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

Find the limit

Explanation:

### Example Question #58 : L'hospital's Rule

Use L'Hospital's rule to evaluate

.

The limit does not exist.

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

The limit does not exist.

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

.

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)

.