Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.8 Continuity - 1.8 Exercises: 27

Answer

continuous on $(-\infty, \sqrt[3]{2})\cup(\sqrt[3]{2}, \infty)$

Work Step by Step

Let $v(x)=x-2$ It is continuous everywhere, by theorem 7 (polynomials, rational functions, root functions, trigonometric functions are are continuous on their domains) Let $u(x)=\sqrt[3]{x}. $ It is continuous everywhere, by theorem 7. Let$ f(x)=u(v(x))=\sqrt[3]{x-2}.$ It is continuous everywhere, by theorem 9 (composite function) Let $g(x)=x^{3}-2$, a polynomial, continuous everywhere, by theorem 7. $Q(x)=\displaystyle \frac{f(x)}{g(x)}$ has domain:$ g(x)\neq 0,$ $x\neq\sqrt[3]{2}$ Domain=$(-\infty, \sqrt[3]{2})\cup(\sqrt[3]{2}, \infty)$ It is continuous on its domain by Th.4.5 (If $f$ and $g$ are continuous, then 5. $\displaystyle \frac{f}{g}$ if $g(a)\neq 0$ is continuous on its domain.
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