   Chapter 2, Problem 7RQ ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 → 5 f ( x ) = 0   and   lim x → 5 g ( x ) = 0 ,   then   lim x → 5 [ f ( x ) / g ( x ) ] does not exist.

To determine

Whether the statement, “if limx5f(x)=0 and limx5g(x)=0, then limx5[f(x)g(x)] does not exist” is true or false.

Explanation

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

Note:

The limit may be infinite or it may be some finite value when both the numerator and the denominator approach 0.”

Reason:

By note 1, the limx5[f(x)g(x)] may be exists.

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

Let f(x)=x225  and g(x)=x5.

This follows that, limx5(x225)=0 and limx5(x5)=0

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