Textbook Problem

The signum (or sign)function, denoted by sgn, is defined by

$\begin{array}{l}\mathrm{sgn}x=\{\begin{array}{l}-1\text{if}x\text{<0}\\ \text{0}\text{if}x=0\\ 1\text{if}x>0\end{array}\\ \end{array}$

(a) Sketch the graph of this function.

(b) Find each of the following limits or explain why it does not exist.

(i) $\underset{x\to 0^{+}}{\lim }\mathrm{sgn}x$

(ii) $\underset{x\to 0^{-}}{\lim }\mathrm{sgn}x$

(iii) $\underset{x\to 0}{\lim }\mathrm{sgn}x$

(iv) $\underset{x\to 0}{\lim }\left|\mathrm{sgn}x\right|$