Evaluating $\lim_{ x \to+ \infty }\left[\frac{[x]}{x}\right]$ and $\lim_{ x \to- \infty }\left[\frac{[x]}{x}\right]$

Question

Find the limit : $$\lim_{ x \to+ \infty }\left[\frac{[x]}{x}\right]=? \\\lim_{ x \to- \infty }\left[\frac{[x]}{x}\right]=?$$

$[x]:$ floor function

I tried:

$[u]∼u :\text{where} x→∞$

$$\lim_{ x \to+ \infty }\left[\frac{[x]}{x}\right]=\lim_{ x \to+ \infty }\frac{[x]}{x}=1!!!$$

Answer 1

For integer $x \ne 0$, we have $[[x]/x] =1$.

For noninteger $x$ such that $|x|>1$, we have $[[x]/x] = \begin{cases}0 & x>1 \\ 1 & x < -1.\end{cases}$

So, $\lim_{x \to \infty} [[x]/x]$ does not exist while $\lim_{x \to -\infty} [[x]/x]=1$.