University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Practice Exercises - Page 111: 40

Answer

$k(x)$ does not have a continuous extension at $x=0$. However, $k(x)$ can still be extended to be continuous from either the left or the right at $x=0$. - To be continuous from the left, $k(x)$ needs to include the value $k(0)=-1.443$. - To be continuous from the right, $k(x)$ needs to include the value $k(0)=1.443$.

Work Step by Step

$$k(x)=\frac{x}{1-2^{|x|}}$$ The graph is enclosed below. Condition to have a continuous extension for $k(x)$ at $x=c$: $\lim_{x\to c}k(x)$ must exist. But here, $\lim_{x\to0}k(x)$ does not exist, because as $x\to0$ from the left and the right, $k(x)$ approaches different values: - As $x\to0^+$, $k(x)$ approaches $1.443$. - As $x\to0^-$, $k(x)$ approaches $-1.443$. So $k(x)$ does not have a continuous extension at $x=0$. However, $k(x)$ can still be extended to be continuous from either the left or the right at $x=0$. - To be continuous from the left, $k(x)$ needs to include the value $k(0)=-1.443$. - To be continuous from the right, $k(x)$ needs to include the value $k(0)=1.443$.
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