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: 18

Answer

$$ f(x)=\frac{5}{x}+\frac{2 x}{x+4} $$ is continuous for all real numbers $x$ except $x = 0 $, $x = -4 $.

Work Step by Step

$$ f(x)=\frac{5}{x}+\frac{2 x}{x+4} $$ suppose $$ f(x)=g(x)+h(x) $$ such that $$g(x)= \frac{5}{x} , \quad \quad h(x)=\frac{2 x}{x+4} $$ Solving the equation $$ x+4=0 $$ yields discontinuities at $x = -4$. Therefore the function $$ h(x)=\frac{2 x}{x+4} $$ is continuous for all real number $x$ except $ x = -4$ Thus $$ f(x)=\frac{5}{x}+\frac{2 x}{x+4} $$ is continuous for all real numbers $x$ except $x = 0 $, $x = -4 $. Finally, values of $x$, at which $f $ is not continuous are $x = 0$, $x = -4$.
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