Find the limit $\lim_{x\to 0} x\left(\left[\frac{1}{x}\right] +\left[\frac{2}{x}\right] +\cdots \left[\frac{10}{x}\right] \right)$ Can someone help me finding the following limit 
$$ \lim_{x\to 0} x\left(\left\lfloor\frac{1}{x}\right\rfloor +\left\lfloor\frac{2}{x}\right\rfloor +\cdots \left\lfloor\frac{10}{x}\right\rfloor\right)$$
I can somehow guess the limit will be $55$, as $\lim_{x\to 0}x\left\lfloor\frac{1}{x}\right\rfloor=1$. But, I am not able to prove it.
Note: $\left\lfloor x\right\rfloor$ denotes the greatest integer less than or equal to $x$.

We can convert to
$$\lim_{t\to\pm\infty}\frac{\lfloor t\rfloor+\lfloor2t\rfloor+\lfloor3t\rfloor+\cdots\lfloor10t\rfloor}t=\\
\lim_{t\to\pm\infty}\left(1+2+3+\cdots10-\frac{\{t\}+\{2t\}+\{3t\}+\cdots\{10t\}}t\right).$$
The second term vanishes because the numerator remains in range $[0,10)$.

Notation: $t=\lfloor t\rfloor+\{t\}$.
