University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.5 - Continuity - Exercises - Page 95: 50

Answer

$f(x)$ can chooses to extend to be either continuous from the left or from the right at the origin: - From the left: the extended function's value should be $-2.3$. - From the right: the extended function's value should be $2.3$.
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Work Step by Step

$$f(x)=\frac{10^{|x|}-1}{x}$$ The graph of the function is shown below. For $f$ to have a continuous extension to the origin, $\lim_{x\to0}f(x)$ must exist first. Yet, looking at the graph, $f(x)$ approaches a value as $x\to0^+$, and $f(x)$ approaches another value as $x\to0^-$. So $\lim_{x\to0}f(x)$ does not exist, and $f$ does not have a continuous extension to the origin. But since $\lim_{x\to0^+}f(x)$ and $\lim_{x\to0^-}f(x)$ do exist, $f(x)$ can still be extended to be continuous at the origin from either the left or the right, if $f(0)$ acquires the value of either $\lim_{x\to0^+}f(x)$ or $\lim_{x\to0^-}f(x)$ - To be continuous at the origin from the left: Zooming in the graph, we see that $\lim_{x\to0^-}f(x)=-2.3$. So $f(0)$ needs to acquire the value $-2.3$. - To be continuous at the origin from the right: Zooming in the graph, we see that $\lim_{x\to0^+}f(x)=2.3$. So $f(0)$ needs to acquire the value $2.3$.
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