Evaluate the limit $\lim\limits_{n\to0}\frac{(x)+(2x)+\cdots (nx)}{n^2}$

This was too long for a comment, hopefully this helps:

Using $1+2+\dots+n=\frac{n(n+1)}{2}$ we find that: \begin{align*} \lim_{n\rightarrow\infty} \frac{(x)+(2x)+\dots+(nx)}{n^2} &=\lim_{n\rightarrow\infty} \frac{x-[x]+2x-[2x]+\dots+nx-[nx]}{n^2}\\ &=\lim_{n\rightarrow\infty} \frac{x(1+2+\dots+n)}{n^2}-\lim_{n\rightarrow\infty} \frac{[x]+[2x]+\dots+[nx]}{n^2}\\ &=x\lim_{n\rightarrow\infty} \frac{n(n+1)}{2n^2}-\lim_{n\rightarrow\infty} \frac{[x]+[2x]+\dots+[nx]}{n^2}\\ &=\frac x2-\lim_{n\rightarrow\infty} \frac{[x]+[2x]+\dots+[nx]}{n^2}\\ \end{align*}