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 2.3, Problem 29E
To determine

To evaluate: The limit value of the function f(x) as x approaches 4.

Expert Solution

Answer to Problem 29E

The limit value of the function f(x) as x approaches 4 is 7.

Explanation of Solution

Given:

The inequality, 4x9f(x)x24x+7 for x0.

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist, then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Theorem used: The Squeeze Theorem

“If g(x)f(x)h(x) when x is near a (except possibly at a) and limxag(x)=limxah(x)=L then limxaf(x)=L.”

Calculation:

Apply the Squeeze Theorem and obtain a function g smaller than f(x) and a function h bigger than f(x) such that both g(x) and h(x) approaches 4.

The given inequality becomes, 4x9f(x)x24x+7.

When the limit x approaches to 4, the inequality becomes,

limx44x9limx4f(x)limx4x24x+7

Let g(x)=4x9 and h(x)=x24x+7.

Obtain the limit value of as x approaches 4 of the function g(x).

limx44x9=limx44xlimx49 (by limit law 2)=4limx4xlimx49 (by limit law 3)=4(4)(9) (by limit law 8 and 7)=7

Obtain the limit value of as x approaches 4 of the function h(x).

limx4x24x+7=limx4(x24x)+limx47 (by limit law 1)=limx4(x2)limx44x+limx47 (by limit law 2)=limx4(x2)4limx4x+limx47 (by limit law 3)

                        =(4)24(4)+(7) (by limit law 9,8 and 7)=1616+7=7

If g(x)f(x)h(x) when x approaches 4 and limx44x9=limx4x24x+7=7, then by Squeeze Theorem the limit of the function f(x) is 7.

That is, limx4f(x)=7.

Thus, the limit value of the function f(x) as x approaches 4 is 7.

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!