Chapter 2.3, Problem 48E

Let g(x) =sgn(sinx).

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

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

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

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

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

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

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

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

(c) Sketch a graph of g.