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.

Yes.  As a simple example, I'll work with sequences.  Let $f(n) = (-1)^n$ and $g(n) = (-1)^{n}$.  Then $f(n)g(n) = 1$, so $1 = \lim_{n \to \infty} f(n)g(n)$, but neither of the individual limits exist.
