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: 20

Answer

$$ f(x)=\left|4-\frac{8}{x^{4}+x}\right| $$ values of $x$, at which $f $ is not continuous are $x = 0$, $ x = -1$

Work Step by Step

$$ f(x)=\left|4-\frac{8}{x^{4}+x}\right| $$ suppose $$ f(x)=|g(x)-h(x)| $$ such that $$g(x)= 4, \quad \quad h(x)=\frac{8}{x^{4}+x} $$ Solving the equation $$ x^{4}+x=x(x^{3}+1)=0 $$ yields discontinuities at $x = 0$ and at $ x = -1$. Therefore the function $$ \frac{8}{x^{4}+x} $$ is continuous for all real numbers $x$ except $x = 0$ , $ x = -1$ Thus the function $$ f(x)=\left|4-\frac{8}{x^{4}+x}\right| $$ is continuous for all real numbers $x$ except $x = 0 $, $x = -1$. Finally, values of $x$, at which $f $ is not continuous are $x = 0$, $ x = -1$
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