Find the value of $\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$

Question

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

Answer 1

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