If a and b are positive integers then 

$\underset{x\to 0}{lim} \frac{x}{a}\left[\frac{b}{x}\right]=\underset{x\to 0}{lim} \frac{x}{a}\left[\frac{b}{x}-\left\{\frac{b}{x}\right\}\right]$

({x} is the fractional part of x)

$\begin{array}{l}=\underset{x\to 0}{lim} \left(\frac{b}{a}-\frac{b}{a}\frac{\{b/x\}}{b/x}\right)\\ =\frac{b}{a}-\frac{b}{a}\underset{y\to \mathrm{\infty }}{lim} \frac{\left\{y\right\}}{y}\\ =\frac{b}{a}\end{array}$

Since $0\le \left\{y\right\}<1$ so $\frac{\left\{y\right\}}{y}\le \frac{1}{y}$ for $y>0$

Hence $\underset{y\to \mathrm{\infty }}{lim} \frac{\left\{y\right\}}{y}=0$