Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.4 - One-Sided Limits - Exercises 2.4 - Page 74: 9

Answer

a. Domain $[0,2]$, Range $y\in(1,1]$ and $y=2$ b. $(0,1)\cup(1,2)$ c. $x=2$ d. $x=0$
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Work Step by Step

We are given the piecewise function: $f(x)=\begin{cases} \sqrt {1-x^2}\hspace0.8cm 0\leq x\lt1 \\ 1\hspace2cm 1\leq x\lt2 \\2\hspace2cm x=2 \end{cases}$, We can graph the function as shown. a. Domain $[0,2]$, Range $y\in(1,1]$ and $y=2$ b. Check the end points; we see that limits do not exist at $x=0,1,2$, thus $\lim_{x\to c}f(x)$ exist for points in $(0,1)\cup(1,2)$ c. For the left-hand limit only, we see a point at $x=2$ with $\lim_{x\to 2^-}f(x)=1$ d. For the right-hand limit only, we see a point at $x=0$ with $\lim_{x\to 0^+}f(x)=1$
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