Answer
(a) (i) $1$; (ii) $-1$; (iii) $DNE$; (iv) $-1$; (v) $1$; (vi) $DNE$
(b) $a=k\pi$, where $k$ is integer
(c) see graph
Work Step by Step
Recall the sign function:
$$\begin{align*}
sgn(x)=\begin{cases}
-1\text{ for }x<0,\\
0,\text{ for }x=0\\
+1\text{ for }x>0
\end{cases}
\end{align*}$$
$g(x)=sgn(\sin(x))$
(a) Evaluate the limits:
(i)$\lim\limits_{x \to 0^{+}}g(x)=1$
because as $x$ approaches $0$ from the right, $\sin(x)$ gets closer to but always greater than $0$, so the sign is positive and the limit is positive $1$.
(ii)$\lim\limits_{x \to 0^{-}}g(x)=-1$
because as x approaches $0$ from the left, $\sin(x)$ gets closer to but always less than $0$, so the sign is negative and the limit is negative $1$
(iii)$\lim\limits_{x \to 0}g(x)=DNE$
because the left and right limits are not the same.
(iv)$\lim\limits_{x \to \pi^{+}}g(x)=-1$
because as $x$ approaches $\pi$ from the right, $\sin(x)$ gets closer to but always less than $0$, so the sign is negative and the limit is negative $1$
(v)$\lim\limits_{x \to \pi^{-}}g(x)=1$
because as $x$ approaches $\pi$ from the left, $\sin(x)$ gets closer to but always greater than $0$, so the sign is positive and the limit is positive $1$
(vi)$\lim\limits_{x \to \pi}g(x)=DNE$
because the left and right limits are not the same.
(b)
$\lim\limits_{x \to a}g(x)=DNE$ when a is a multiple of $\pi$
(c) see graph
