Answer
a. Graph for sgn(x) is shown
b. (i) 1 (ii) -1 (iii) DNE (iv) 1
Work Step by Step
1. $x\rightarrow 0^{+}$ from right of 0, sgn x=1
$\lim _{x\to 0^+}\left(sgn\:x\right)=1$
2. $x\rightarrow 0^{-}$ from left of 0, sgn x=-1
$\lim _{x\to 0^+}\left(sgn\:x\right)=-1$
3. As we know from (1) and (2) $\lim _{x\to 0^+}\left(sgn\:x\right)=1 \ne \lim _{x\to 0^+}\left(sgn\:x\right)=-1$
Since the two sides do not equal, limit does not exist.
$\lim _{x\to 0}\left(sgn\:x\right)=D.N.E.$
4.
$\lim _{x\to 0^+}\left(\left|sgnx\right|\right)=1$
$\lim _{x\to 0^-}\left(\left|sgnx\right|\right)=1$
Since two sides are equal then
$\lim _{x\to 0}\left(\left|sgnx\right|\right)=1$
