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: 56


The statement “f and g are both continuous at $ a $, although $ f+g $ is not” does not make 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 $ f+g $ is continuous at a, or not. Find the value of $\left( f+g \right)\left( x \right)$ at $ a $, $\left( f+g \right)\left( a \right)=f\left( a \right)+g\left( a \right)$ Since, $ f\left( a \right)$ and $ g\left( a \right)$ are defined, so is $\left( f+g \right)\left( a \right)$. Now find the value of $\,\underset{x\to a}{\mathop{\lim }}\,\left( f+g \right)\left( x \right)$, $\,\underset{x\to a}{\mathop{\lim }}\,\left( f+g \right)\left( x \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$. Since, $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ and $\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$ both exist, thus $\,\underset{x\to a}{\mathop{\lim }}\,\left( f+g \right)\left( x \right)$ also exists. From the above steps, $\,\underset{x\to a}{\mathop{\lim }}\,\left( f+g \right)\left( x \right)=f\left( a \right)+g\left( a \right)=\left( f+g \right)\left( a \right)$ Thus, the function satisfies all the properties of being continuous. Thus, the function $ f+g $ is continuous at a. Hence, the statement does not make sense.
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