Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Review - Concept Check: 8


(a) $f(x)=x$, $f(x)=e^{x}$, $f(x)=x^3+2x-1$ (b) $$f(x)=\left\{_{x,\quad x\in[0,1/2]}^{2x,\quad x\in(1/2,1],}\right.$$

Work Step by Step

(a) The examples are any of the elementary functions that include $[-1,1]$ in their domains. These are $f(x)=x$, $f(x)=e^{x}$, $f(x)=x^3+2x-1$ and so on. (b) An example can be the function $$f(x)=\left\{_{x,\quad x\in[0,1/2]}^{2x,\quad x\in(1/2,1],}\right.$$ The left sided limit at $x=1/2$ is $$\lim_{x\to\frac{1}{2}^-}f(x)=\lim_{x\to\frac{1}{2}^-}x=\frac{1}{2}.$$ The right sided limit at $x=1/2$ is $$\lim_{x\to\frac{1}{2}^+}f(x)=\lim_{x\to\frac{1}{2}^+}2x=2\cdot\frac{1}{2}=1.$$ They are obviously different so this function is discontinuous at $x=1/2$.
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