Limit of function involving floor function

Question

What would be $$ \lim_{x\to o}\sum_{i=1}^{10}x[i/x] $$ using the fact that $\lim_{x\to 0} x[1/x]=1$. Is the limit in question $55$?

Answer 1

$$x\left\lfloor\frac ix\right\rfloor=i-x\left\{\frac ix\right\}.$$

As the fractional parts are bounded, the second term vanishes in the limit and

$$\lim_{x\to0}\sum_{i=1}^{10}x\left\lfloor\frac ix\right\rfloor=\sum_{i=1}^{10}i.$$

Answer 2

Let $\dfrac1{n+1}

$\sum \dfrac n{n+1}*i

Then Left and Right converge to $\sum i$.