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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 4, Problem 7RCC

**(a)**

To determine

**To state:** The L’Hospitals rule and explain.

Expert Solution

**The L’Hospital’s rule:**

“Suppose *f* and *g* are differentiable functions with *a* such that

It states that if a limit of a quotient of a function is in an indeterminate form, then the limit of the quotient of the function is same as the limit of quotient of the derivatives subject to condition that all limits exist.

**(b)**

To determine

**To explain:** The use of L’Hospital’s rule for the limit of product of the functions *fg* such that

Expert Solution

**Quotient rule for limits:** “Limit of the quotient is quotient of the limits”.

**Calculation:**

Rearrange the given function in quotient form and then use the Quotient rule.

On further simplification,

So on applying L’Hospitals rule, it is easy to evaluate the given limit.

**(c)**

To determine

**To explain:** The use of L’Hospital’s rule for the limit of difference of the functions *f**g* such that

Expert Solution

**Sum rule for limits:** “Limit of the sum is sum of the limits”.

**Calculation:**

Rewrite the given function as follows.

Use the sum rule and find the limit as follows,

On further simplification,

So on applying L’Hospitals rule, it is easy to evaluate the given limit.

**(d)**

To determine

**To explain:** The use of L’Hospital’s rule for

Expert Solution

**Calculation:**

Find the limit as follows.

The value of the above limit is in an indeterminate form.

So on applying L’Hospitals rule, it is easy to evaluate the given limit.