TITLE: Compute $\lim_{x\to 0}\frac{x}{[x]}$
QUESTION [7 upvotes]: When I take left hand limit of the function $\lim\limits_{x\to 0}\frac{x}{[x]}$, then $\lim\limits_{h\to 0^{-}}\frac{-h}{[-h]}=\lim_{h\to 0^{-}}\frac{-h}{-1}=0$ where $0<h<1$ and $[\cdot ]$ is greatest integer function. But when I take right hand limit, then the function $\frac{h}{[h]}$ does not exists. so I do not understand what about $\lim\limits_{h\to 0^{+}}\frac{h}{[h]}$ and the limit of original function. Does the limit exists? please someone help me
Thanks in advance.

REPLY [2 votes]: There  is a   theorem in Analysis which states:

Given a limit point $c$  of   a set   $A$    and  a  function      $f:A\rightarrow\mathbb{R}$,  $\lim_{x\rightarrow c}f(x)=L$     if and only if 
  \begin{align*}
\lim_{x\rightarrow c^-}f(x)=L\qquad\text{and}\qquad\lim_{x\rightarrow c^+}f(x)=L
\end{align*}

Since in our example the right-hand limit $ \lim_{x\rightarrow 0^+}\frac{x}{[x]}$  does not exist it follows, that the limit  $$\lim_{x\rightarrow 0}\frac{x}{[x]}$$ does not exist.