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

$$\mathrm{sgn}\; {x} = \begin{cases} -1 & \textrm{if }x < 0 \\ 0 & \textrm{if }x = 0 \\ 1 & \textrm{if }x > 0\\ \end{cases}$$

Find each of the following limits.If the limit does not exist, enter DNE below.

(a) $\displaystyle \lim_{x \to 0^+} \mathrm{sgn} \;{x}$ =
(b) $\displaystyle \lim_{x \to 0^-} \mathrm{sgn}\; {x}$ =
(c) $\displaystyle \lim_{x \to 0} \mathrm{sgn} \;{x}$ =
(d) $\displaystyle \lim_{x \to 0} |\mathrm{sgn}\; {x}|$ =