Find the value of $\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$ 
Determine if the following limits exist
$$\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$$

note that $$\frac{1}{x}-1 <\lfloor  \frac{1}{x}\rfloor  \leq \frac{1}{x}$$
$$1-x <x\lfloor  \frac{1}{x}\rfloor  \leq 1$$
i'm stuck here

Observe that $\lfloor\frac1x\rfloor=0$ for $x>1$, hence $x\lfloor\frac1x\rfloor$ is identically zero on $]1,+\infty[$. Hence the limit is $0$.
