Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 1 - Limits and Continuity - 1.5 Continuity - Exercises Set 1.5 - Page 99: 17

Answer

$$ f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1} $$ values of $x$, at which $f $ is not continuous are $x = 0$, $x = 1$ and $ x = -1$

Work Step by Step

$$ f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1} $$ suppose $$ f(x)=g(x)+h(x) $$ such that $$g(x)= \frac{3}{x}, \quad \quad h(x)=\frac{x-1}{x^{2}-1}$$ The function $ g(x)=\frac{3}{x}$ being graphed is a rational function, and hence is continuous at every number except $ x=0$. Now consider the other function $$ x^{2}-1=0 $$ yields discontinuities at $x = 1$ and at $ x = -1$. Therefore the function $$ h(x)=\frac{x-1}{x^{2}-1} $$ is continuous for all real numbers $x$ except $x = 1$ , $ x = -1$ Thus $$ f(x)=\frac{3}{x}+\frac{x-1}{x^{2}-1} $$ is continuous for all real numbers $x$ except $x = 0 $, $x = 1$ and $ x = -1$.
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