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 $0$.


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