Finding the value of one-sided limits and greatest integer function. 
$$
\lim_{x \to 0} \frac{a}{x} \left\lfloor\frac{x}{b} \right\rfloor
$$

The $\lfloor \rfloor$ stands for the greatest integer function.
I have calculated and the left-hand limit is coming as (ab). But, I have doubt in the right-hand limit. I did this problem by sandwich-theorem. Can, anyone help me to find the right-hand limit correctly?

I assume that
$a > 0$
and $b > 0$.
If $0 < x < b$,
$\lfloor \frac{x}{b} \rfloor
= 0$.
Therefore,
for
$0 < x < b$,
$\frac{a}{x}\lfloor \frac{x}{b} \rfloor
= 0$.
Therefore,
$\lim_{x \to 0^{+}}\frac{a}{x}\lfloor \frac{x}{b} \rfloor
= 0$.
The expression is not defined at $x = 0$.
If $0 > x > -b$,
$\lfloor \frac{x}{b} \rfloor
= -1$.
Therefore,
for
$0 > x > -b$,
$\frac{a}{x}\lfloor \frac{x}{b} \rfloor
= -\frac{a}{x}$.
This is not defined
as $x \to 0^{-}$.
