Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.4 Limits and Continuity - Exercises - Page 68: 82


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Work Step by Step

We are given: $f(x)=|x|$ Rewrite the function: $f(x)=\begin{cases} -x,\text{ for }x\leq 0\\ x,\text{ for }x>0 \end{cases}$ Compute the left hand and right hand limits: $\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=\displaystyle\lim_{x\rightarrow 0^{-}} (-x)=-0=0$ $\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=\displaystyle\lim_{x\rightarrow 0^{+}} x=0$ As $\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=f(0)=0$, the function is continuous in $x=0$. The function is also continuous on $(-\infty,0)$ and $(0,\infty)$,therefore it is continuous on $\mathbb{R}$.
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