Chapter 1.6, Problem 48E

Let. $g\left(x\right)=\mathrm{sgn}\left(\sin x\right)$.

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

(i) $\underset{x\to 0^{+}}{\lim }g\left(x\right)$

(ii) $\underset{x\to 0^{-}}{\lim }g\left(x\right)$

(iii) $\underset{x\to 0}{\lim }g\left(x\right)$

(iv) $\underset{x\to {\pi }^{+}}{\lim }g\left(x\right)$

(v) $\underset{x\to {\pi }^{-}}{\lim }g\left(x\right)$

(vi) $\underset{x\to \pi }{\lim }g\left(x\right)$

(b) For which values of a does ${{\displaystyle \lim }}_{x\to a}g\left(x\right)$ not exist?

(c) Sketch a graph of g.