Compute $\lim_{x\to 0}\frac{x}{[x]}$

Question

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.

Answer 1

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.