Find $\lim_\limits{x\to 0}{x\left[1\over x\right]}$.
Attempt: $\lim_\limits{x\to 0}{x\left[1\over x\right]}=\lim_\limits{x\to 0}x\left ({1\over x}-\{{1\over x}\}\right)=\lim_\limits{x\to 0}\left (1-{x\{{1\over x}\}}\right).$
Since $\{{1\over x}\}$ is bounded and $x\to 0$, then $x\cdot\{{1\over x}\}\to 0$ and therefore $\lim_\limits{x\to 0}{x\left[1\over x\right]}=1.$
Another approach using sandwich: $$x\left(\frac1x-1\right)<x\left[\frac1x\right]\le x\frac1x$$