Question about limit of a product 
Is it possible that both $\displaystyle\lim_{x\to a}f(x)$ and $\displaystyle\lim_{x\to a}g(x)$ do not exist but $\displaystyle\lim_{x\to a}f(x)g(x)$ does exist?

The reason I ask is that I was able to show that if $\displaystyle\lim_{x\to a}f(x)$ does not exist but both $\displaystyle\lim_{x\to a}g(x)$ and $\displaystyle\lim_{x\to a}f(x)g(x)$ do exist, then $\displaystyle\lim_{x\to a}g(x)=0$. However, I'm not sure whether the assumption on the existence of $\displaystyle\lim_{x\to a}g(x)$ is necessary.

If you have two disjoint sequences $u_n\to a$ and $v_n\to a$
i.e. $\{a_n\mid n\in\mathbb N\}\cap\{b_n\mid n\in\mathbb N\}=\varnothing$
You can define $\begin{cases}f(a_n)=1 & \forall n\in\mathbb N\\f(x)=0&\text{otherwise}\end{cases}\quad$ and $\quad\begin{cases}g(b_n)=1 & \forall n\in\mathbb N\\g(x)=0&\text{otherwise}\end{cases}$
It is clear that $f(x)g(x)=0$ everywhere so it has a limit zero, while neither $f$ nor $g$ has a limit. And you can smoothen these functions if desired to get a $C^\infty$ example, just make sure that $f$ and $g$ bumps do not coincide.
