Why is true? $\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}$ $$\begin{array}{l}a,b > 0\\\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}\\\end{array}
$$
I asked already a similar question, but I'm still not sure what makes it true.
As $x$ is decreasing to $0$, $x \over a$ is converging to $0$ but $\left[ {\frac{b}{x}} \right]$ is converging to $\infty$, So we left with $0*\infty$ which isn't helpful.  
How can you solve it?

Hint
Use the squeeze theorem knowing
$$\frac{b}{x}-1\le\left[ {\frac{b}{x}} \right]\le \frac{b}{x}$$
