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