# 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?

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$0 \le \frac{x}{a} \lceil \frac{b}{x} \rceil \le \frac {x}{a} \frac{b}{x}$ –  user45878 Dec 17 '13 at 20:38
Are you implying the limit is $0$? –  AndrePoole Dec 17 '13 at 20:40

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