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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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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