real analysis - $limlimits_{x to 0^+}f(x)$ and $limlimits_{x to 0^+}g(x)$ do not exist, $limlimits_{x to 0^+}f(x)g(x)$ does

I am looking for functions $f, g: (0, +\infty) \to (0, +\infty)$ such that $\lim\limits_{x \to 0^+}f(x)$ and $\lim\limits_{x \to 0^+}g(x)$ do not exist, however, $\lim\limits_{x \to 0^+}f(x)g(x)$ does. So far, I have only encountered two functions for which the one-sided limit doesn't exist: $\sin{1 \over x}$ and $\cos{1 \over x}$. This doesn't seem to be what I'm looking for. I would appreciate any hints.

Answer

For example:

$$f(x)=\begin{cases} 2 & \text{if}& x\text{ rational}\\1/2 & \text{if}& x\text{ irrational},\end{cases}\qquad g(x)=\begin{cases} 2 & \text{if}& x\text{ irrational}\\1/2 & \text{if}& x\text{ rational.}\end{cases}$$