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 2.5 - Continuity - Exercises 2.5 - Page 84: 5


$a.\quad $yes $b.\quad $yes $c.\quad $yes $d.\quad $yes

Work Step by Step

The graph of f(x) contains points $(x,f(x))$, where x belongs to the domain of f. $a.$ The point $(-1,0)$ belongs to the graph of f, so $f(-1)=0$ ( $f(-1)$ exists) $b.$ As x approaches the value $-1$ from the right, f(x) approaches $0$. The right-sided limit exists, $\displaystyle \lim_{x\rightarrow 1^{+}}f(x)=0$, $c.$ $\displaystyle \lim_{x\rightarrow 1^{+}}f(x)=0$= $f(-1)$ (they are equal) $d.$ The domain of f contains the half-closed interval $[-1,0)$. The result of (c) implies that f is right-continuous at the left (closed) border, (see the first definitions of the section, and the discussion below them, referring to continuity over closed and half-closed intervals). So we say that f is continuous at $x=-1$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.