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, Problem 3RQ
To determine

Whether the statement, limx1(x3x2+2x4)=limx1(x3)limx1(x2+2x4) is true or false.

Expert Solution

Answer to Problem 3RQ

The statement is true.

Explanation of Solution

Quotient Law: Suppose that the limits limxaf(x) and limxag(x) exist.

Then limxaf(x)g(x)=limxaf(x)limxag(x) if limxag(x)0

Reason:

Let the function Q(x)=f(x)g(x) where f(x)=x3 and g(x)=x2+2x4.

By the Quotient Law, the given statement is true only if the individual limits exist and the limit of the denominator Q(x) is not equal to zero as x approaches 1.

That is, limx1(x3x2+2x4)=limx1(x3)limx1(x2+2x4) is true whenever limx1(x3) and limx1(x2+2x4) exists and limx1(x2+2x4) is not equal to zero.

The limit f(x)=x3 as x approaches 1 is computed as follows,

limx1(x3)=13=2

Thus, the limit exists.

Compute the limit g(x)=x2+2x4 when x approaches 1.

limx1(x2+2x4)=(1)2+2(1)4=1+24=34=1

Thus, the limit exists and it is not equal to zero.

Since the individual limit exist and the denominator g(x)=x2+5x6 is not equal to zero as x approaches 1, the Quotient law is applicable here.

Therefore, the statement is true.

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