Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 1 - Functions and Limits - 1.8 Continuity - 1.8 Exercises: 3

Answer

a. f is discontinuous at $-4,-2,2,$ and $4$ b. At $-4$, from neither side At $-2$, from the left At $2$, from the right At $4$, from the right

Work Step by Step

A function $f$ is continuous at a number $a$ if $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$. A function $f$ is continuous from the right at a number $a$ if $\displaystyle \lim_{x\rightarrow a^{+}}f(x)=f(a)$ and $f$ is continuous from the left at $a$ if $\displaystyle \lim_{x\rightarrow a^{-}}f(x)=f(a)$ --------------- a. $f(-4)$ is not defined, At $x=-2,\ 2$, and $4$, the left and right limits are not equal (so the limits do not exist). b. At $-4$, $f$ is not continuous from either side since $f(-4)$ is not defined. At $-2$, $f$ is continuous from the left since $\displaystyle \lim_{x\rightarrow-2^{-}}f(x)=f(-2)$. At $2,\ f$ is continuous from the right since $\displaystyle \lim_{x\rightarrow 2^{+}}f(x)=f(2)$. At $4,\ f$ is continuous from the right since $\displaystyle \lim_{x\rightarrow 4^{+}}f(x)=f(4)$
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