Limit of a greatest integer function (sided limit)

Question

What is the value of $\lim\limits_{x\to 0^+} \dfrac{b}{x}\left\lfloor\dfrac{x}{a}\right\rfloor$ for $a>0$ and $b>0$. Note that $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.

I know that $\left\lfloor\dfrac xa\right\rfloor=0$. But when it comes to $\lim\limits_{x\to0^+}\dfrac bx\cdot0$, the result is indeterminate. I want to know how to remove this indetermination.

Answer 1

For sufficiently small but positive $x$ (in particular $0<x<a$), we have that $0<x/a<1$. This means that $\lfloor x/a\rfloor= 0$ so that the limit is just $\lim_{x\to 0^+}(b/x)\cdot0=0$.

Note that the requirement that $x$ can't be $0$ comes from the definition of a limit-- $x$ gets closer and closer to, but never equals, $0$.

Answer 2

Since $\frac{b}{x}\left \lfloor{\frac{x}{a}}\right \rfloor=0,\forall x\in (0,a)$, it follows that $\lim_{x\rightarrow 0^+}\frac{b}{x}\left \lfloor{\frac{x}{a}}\right \rfloor=0$ .