   Chapter 1, Problem 12RQ

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.12. If limx→5 f(x) = 2 and limx→5 g(x) = 0, then limx→5 [f(x)/g(x)] does not exist.

To determine
Whether the statement “if limx5f(x)=2 and limx5g(x)=0, then limx5[f(x)g(x)] does not exist is true or false” and also give explanation.

Explanation

The statement is to be proved in four cases.

Case 1:

Assume limx5g(x)=0+=limx5+g(x).

Solve the equation limx5[f(x)g(x)] as follows.

limx5[f(x)g(x)]=limx5f(x)limx5g(x)=20+=

Therefore, the limit function limx5[f(x)g(x)] does not exist when limx5g(x) is zero from the right.

Case 2:

Assume limx5g(x)=0=limx5+g(x).

Solve the equation limx5[f(x)g(x)] as follows.

limx5[f(x)g(x)]=limx5f(x)limx5g(x)=20=

Therefore, the limit function limx5[f(x)g(x)] does not exist when limx5g(x) is zero from the left.

Case 3:

Assume limx5g(x)=0,limx5+g(x)=0+.

Solve the equation limx5[f(x)g(x)] as follows.

limx5[f(x)g(x)]=limx5f(x)limx5g(x)=20=

Solve the equation limx5+[f(x)g(x)] as follows

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