Can the limit of a product exist if neither of its factors exist? Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.
Sorry if this seems elementary, I have just started my degree...
Thanks in advance.

I would like to reiterate a solution given above. Neither of the limits $$\lim_{x\to0}\left(\sin{\frac1x}\right), ~~~\lim_{x\to0}\left(\frac{1}{\sin{\frac1x}}\right)$$ exist, but $$\lim_{x\to0}\left(\sin{\frac1x}\right)\left(\frac{1}{\sin{\frac1x}}\right)=1.$$
I am aware that $ \large f(x) = \frac{1}{\sin{\frac1x}}$ has infinite discontinuities as x approaches zero. In particular the expression is undefined when $  f(\frac{1}{\pi n }) =\large \frac{1}{\sin{\frac{1}{ \left( \frac{1}{ \pi n } \right) }}} $. But I don't see why this is a problem as far as the limit goes. 
