BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4, Problem 7RCC

(a)

To determine

To state: The L’Hospitals rule and explain.

Expert Solution

Explanation of Solution

The L’Hospital’s rule:

“Suppose f and g are differentiable functions with g(x)=0 on an open interval that contains a such that limxaf(x)=0 and limxag(x)=0 or limxaf(x)= and limxag(x)= then limxaf(x)g(x)=limxaf(x)g(x)”.

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 f0 and g as xa.

Expert Solution

Explanation of 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.

limxa[f(x)g(x)]=limxaf(x)1g(x)=limxaf(x)limxa1g(x)=limxaf(x)1limxag(x)=01

On further simplification, limxa[f(x)g(x)]=00, which is in an indeterminate form.

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 fg such that f and g as xa.

Expert Solution

Explanation of Solution

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

Calculation:

Rewrite the given function as follows.

f(x)g(x)=(1g(x)1f(x))(f(x)g(x))

Use the sum rule and find the limit as follows,

limxa[f(x)g(x)]=limxa[(1g(x)1f(x))(f(x)g(x))]=limxa(1g(x)1f(x))limxa(f(x)g(x))=(1limxag(x)1limxaf(x))(limxaf(x)limxag(x))=(11)()

On further simplification, limxa[f(x)g(x)]=0, which is in an indeterminate form.

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 [f(x)]g(x) such that f0 and g0 as xa.

Expert Solution

Explanation of Solution

Calculation:

Find the limit as follows.

lnlimxa[f(x)g(x)]=limxaln(f(x)g(x))=limxag(x)[ln(f(x))]=limxaln(f(x))1g(x)=ln(0)10.

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.

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