Calculate $\lim_{x \to 0^+} x\lfloor 1/x\rfloor$

Question

I need to find the following limit: $$ \lim\limits_{x \to 0^+} x \left\lfloor\frac{1}{x}\right\rfloor $$

Thanks!

Answer 1

HINT: Let $n$ be a positive integer. What is $\left\lfloor\dfrac1x\right\rfloor$ when $\dfrac1{n+1}<x\le\dfrac1n$? What happens to $x\left\lfloor\dfrac1x\right\rfloor$ as $x$ ranges from $\dfrac1n$ down to $\dfrac1{n+1}$?

Answer 2

Change $t=\frac{1}{x}$ then you have that $$\lim\limits_{x \to 0^+} x \left\lfloor\frac{1}{x}\right\rfloor=\lim_{t\to\infty}\frac{\lfloor t \rfloor}{t}=1$$ The last one is clear since $t-1\leq\lfloor t \rfloor\leq t+1$