$$\lim_{x\to0} Sgn(x)$$ What should its value be? I know $Sgn(0)=0$, but if we imply that x tends to 0, shouldn't it be an infinitesimal number close to 0, but not equal to zero, and shouldn't its value be 1?
Also, what should $$\lim_{x\to0} Sgn(Sgn(Sgn(x)))$$ be? Shouldn't it too be equal to 1?
Saying the limit is zero, is wrong.
If $\lim\limits_{x\to0+}$ and $\lim\limits_{x\to0-}$ both exist (as finite numbers) and are not equal to each other, then $\lim\limits_{x\to0}$ does not exist.
In some contexts, it might make sense to say it exists as a "principal value", taking an average: $\displaystyle \frac 1 2 \left( \lim_{x\to0+} + \lim_{x\to0-} \right),$ but that is not what is conventionally done when the concept of limit is first introduced, and I would allow is only when the context for it has been explicitly set.