Chapter 1: Limits

Section 1.6: Continuity

Example 1.6.3

$$ Show that the function $f\left(x\right)\=\sin \left(x\right)\/\left|x\right|$ is discontinuous at $x\=0$. $$ Solution $$ •  The graph of $f$ in Figure 1.6.3(a) suggests that this function is discontinuous at $x\=0$ because it has a jump there, that is, the limits from the left and right will be different.   •  Indeed, $\underset{x\to 0-}{lim}\frac{\sin \left(x\right)}{\left|x\right|}$ = $-1$ and $\underset{x\to 0\+}{lim}\frac{\sin \left(x\right)}{\left|x\right|}$ = $1$    •  Hence, $\underset{x\to 0}{lim}f\left(x\right)$ does not exist, so $f$ cannot be continuous at $x\=0$. Figure 1.6.3(a)   Graph of $\sin \left(x\right)\/\left|x\right|$ $$

<< Previous Example   Section 1.6    Next Example >>

© Maplesoft, a division of Waterloo Maple Inc., 2024. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.

For more information on Maplesoft products and services, visit www.maplesoft.com