One sided limit and its value Use the concept of one-sided limits to show that f (x) = x[[x]] has a limit as x approaches 0. What is the value of the limit?

As $x$ approaches zero from the left, $x \to 0$, and $\lfloor{x}\rfloor \to -1$, hence
$$\lim_{x\to 0^{-}} x\lfloor{x}\rfloor = (0)(-1) = 0$$
As $x$ approaches zero from the right, $x \to 0$, and $\lfloor{x}\rfloor \to 0$, hence
$$\lim_{x\to 0^{+}} x\lfloor{x}\rfloor = (0)(0) = 0$$

It follows that ${\displaystyle{\lim_{x\to 0} x\lfloor{x}\rfloor = 0}}$.
