$\displaystyle lim_{x \to 0^{+}} \frac{b}{x}*[\frac{x}{a}] =$
(a > 0 , b > 0)
Which is the greatest integer function or the floor function.
We know that $\displaystyle a>0$ so it follows that $\displaystyle \lim _{x \to 0^ + } \frac{x}{a} = 0$.
That being the case at some point, $\displaystyle \frac{x}{a}<1 $ so that means $\displaystyle \left\lfloor {\frac{x}{a}} \right\rfloor = 0$.
Now what can you say about this limit?
That has absolutely nothing to do with this question.
Do you understand that $\displaystyle \lim _{x \to 0^ + } \frac{x}{a} = 0$ means that $\displaystyle \left( {\exists \delta > 0} \right)\left[ {0 < x < \delta \Rightarrow \quad 0<\frac{x}{a} < 1} \right]$
Which means in turn that $\displaystyle \left( {\exists \delta > 0} \right)\left[ {0 < x < \delta \Rightarrow \quad \left\lfloor {\frac{x}{a}} \right\rfloor = 0} \right]$
Which implies $\displaystyle \left( {\exists \delta > 0} \right)\left[ {0 < x < \delta \Rightarrow \quad \frac{b}{x}\left\lfloor {\frac{x}{a}} \right\rfloor = 0} \right]$.
Note is does not matter what $\displaystyle b$ is.