   Chapter 2, Problem 10RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If lim x → 6 [ f ( x ) g ( x ) ] exists, then the limit must be f(6) g(6).

To determine

Whether the statement, “if limx6[f(x)g(x)] exists, then the limit must be f(6)g(6)” is true or false.

Explanation

Product law: Suppose that the limits limxaf(x) and limxag(x) exists, then

limxa[f(x)g(x)]=limxaf(x)limxag(x).

Calculation:

The given statement is false since the below example disproves the given statement.

Let f(x)={(x6)2if x63if x=6  and g(x)=1(x6)2 and a=6.

It is need to be check, limx6[f(x)g(x)]=f(6)g(6)

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