Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 11 - Section 11.2 - Finding Limits Using Properties of Limits - Exercise Set - Page 1155: 77


The statement "I am working with functions $ f $ and $ g $ for which $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$, $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=0$, and $\underset{x\to 4}{\mathop{\lim }}\,\frac{g\left( x \right)}{f\left( x \right)}\ne 0$ " makes sense.

Work Step by Step

According to the quotient property of limits, $\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)},\text{ }if\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)\ne 0$ But here, $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$, $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=0$ So, the quotient property of limits cannot be applied. Thus, $\underset{x\to 4}{\mathop{\lim }}\,\frac{g\left( x \right)}{f\left( x \right)}\ne 0$ Thus, the given statement makes sense.
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