TITLE: How can I calculate this limit: $\lim\limits_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor$?
QUESTION [10 upvotes]: Calculate the following limit.
$$
\lim_{x\to 0} x\left\lfloor\frac{1}{x}\right\rfloor
$$
Where $\left\lfloor x \right\rfloor$ represents greatest integer function or floor function, i.e greatest integer less than or equal to $x$.
Thanks.

REPLY [2 votes]: Use one basic inequality concerning integer part
$$\frac{1}{x}\leq [\frac{1}{x}]\leq \frac{1}{x}+1$$
\begin{align}
1 \leq x[\frac{1}{x}]\leq 1+x \,\mathrm{if}\, x>0\\
1\geq x[\frac{1}{x}]\geq 1+x \,\mathrm{if}\, x<0
\end{align}
So by squeeze theorem $$\lim _{n\rightarrow 0}x[\frac{1}{x}]=0$$