Limits and Continuity · Superchallenge problem

Evaluate $\displaystyle\lim_{x\to 0^+}x\left(\left\lfloor\dfrac{1}{x}\right\rfloor+\left\lfloor\dfrac{2}{x}\right\rfloor+\cdots+\left\lfloor\dfrac{10}{x}\right\rfloor\right)$, where $\lfloor\,\cdot\,\rfloor$ is the greatest integer function.