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$$\lim_{x\to0} Sgn(x)$$ 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?

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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.

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  • $\begingroup$ No, it depends on which side you come from. Limits from the left yield your result $1$ whereas limits from the right lead to $-1$. Otherwise, sgn would be a continuous function if you replace the value at zero as $1$ which is definitely not the case. $\endgroup$ – YoungMath Aug 15 '18 at 14:46
  • $\begingroup$ @YoungMath didn't you just say the opposite? If approaching from right, sgn yields $1$, and if approaching from the left, sgn yields $-1$. At least that's what I know. $\endgroup$ – Arka Seth Aug 15 '18 at 15:27
  • $\begingroup$ @ArkaSeth, of course, you're right. Sorry. That's the well known difficulty to distinguish left from right. Or was it the other way around? :D $\endgroup$ – YoungMath Aug 15 '18 at 19:21

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