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Find $\lim_{x\rightarrow0}\frac{x}{[x]}$

$[x]$ represent greatest integer less than or equal to x.

Right hand limit is not defined as [0+]=0, left hand limit is zero as [0-]=-1. I want to know whether we can say limit exist or not. Because Left Hand Limit $\ne$Right Hand Limit

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  • $\begingroup$ Going to add a perspective in the comments, because it's not really an answer, and is probably more technical than OP is looking for. I think this could be a matter of definition. I could certainly see saying the limit exists under the epsilon-delta definition if we consider the function on (the closure) of its domain of definition. $\endgroup$ Jun 29, 2018 at 16:35

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For $x\to 0$ the expression $\frac{x}{[x]}$ is not well defined since for $0<x<1$ it corresponds to $\frac x 0$ and thus we can't calculate the limit for that expression.

As you noticed, we can only consider

$$\lim_{x\rightarrow0^-}\frac{x}{[x]}=0$$

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  • $\begingroup$ Can we say at $\lim_{x\rightarrow0^-}$ limit exist $\endgroup$ Jun 29, 2018 at 16:33
  • $\begingroup$ @SamarImamZaidi Yes of course! While RHL is not defined at all since the expression involved is not defined for $0<x<1$. $\endgroup$
    – user
    Jun 29, 2018 at 16:34
  • $\begingroup$ Actually, You answered your own question...the left and right limit are not the same (or one of them is not defined...) so there is not such a "limit" $\endgroup$
    – dmtri
    Jun 29, 2018 at 23:33

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