Limit $\lim\limits_{a \to 0}( a\lfloor\frac{x}{a}\rfloor)$

We know that $\lfloor x/a \rfloor \le x/a < \lfloor x/a \rfloor +1$, so $$ a > 0 \Rightarrow a\lfloor x/a \rfloor \le x < a\lfloor x/a \rfloor +a \\ a < 0 \Rightarrow a\lfloor x/a \rfloor +a < x \le a\lfloor x/a \rfloor < $$ which means that $$ a > 0 \Rightarrow 0 \le x - a\lfloor x/a \rfloor < a \\ a < 0 \Rightarrow a < x - a\lfloor x/a \rfloor \le 0, $$ and so $$ | x - a\lfloor x/a \rfloor | \le |a|. $$ Thus $a \lfloor x/a \rfloor \to x$ as $a \to 0$ by the squeeze lemma.