Question

The value of $\underset{x\to 0}{lim}\mathrm{sgn}\left(x\right)$ is
(where 'sgn' represents signum function)

• A. $0$
• B. $1$
• C. $-1$
• D. Does not exist

Answer 1

The correct option is D Does not exist

Given : $\underset{x\to 0}{lim}\mathrm{sgn}\left(x\right)$
We know that
$\mathrm{sgn}\left(x\right)=\{\begin{array}{cc}-1& x<0\\ 0& x=0\\ 1& x>0\end{array}$

$R.H.L.=\underset{x\to 0^{+}}{lim}\mathrm{sgn}x$
$=\underset{h\to 0}{lim}\mathrm{sgn}(0+h)=1$
$L.H.L.=\underset{x\to 0^{-}}{lim}\mathrm{sgn}\left(x\right)$
$=\underset{h\to 0}{lim}\mathrm{sgn}(0-h)=-1$
Hence, limit does not exist.