Precalculus (6th Edition) Blitzer

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

Chapter 11 - Section 11.3 - Limits and Continuity - Exercise Set - Page 1162: 57


The statement “f and g are both continuous at $ a $, although $\frac{f}{g}$ is not” makes sense.

Work Step by Step

For a function to be continuous at a point a, the function must satisfy the following three conditions: (a) f is defined at a. (b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. (c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ Since, f and g are continuous functions at a, both of them satisfy the above three conditions. Now check whether the function $\frac{f}{g}$ is continuous at a, or not. Find the value of $\left( \frac{f}{g} \right)\left( x \right)$ at $ a $, $\left( \frac{f}{g} \right)\left( a \right)=\frac{f\left( a \right)}{g\left( a \right)}$ If $ g\left( a \right)=0$, then the value $\left( \frac{f}{g} \right)\left( a \right)$ is not defined. In this case, the function $\left( \frac{f}{g} \right)\left( x \right)$ does not satisfy all the conditions of being continuous. Thus, the function $\left( \frac{f}{g} \right)\left( x \right)$ is not continuous at a. Hence, the statement makes sense.
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