Show by means of an example that $\lim_{x \to a} [f(x)g(x)]$ may exist even though neither $\lim_{x \to a} f(x)$ nor $\lim_{x \to a} g(x)$ exists.

Question

This is particularly a hard problem to solve, "a" has to be a defined number. We can't really pick any two functions simply like $\frac{1}{x}$ and any other to exemplify. Please help me with it

Answer 1

Let $f(x)$ be equal to $1$ at rational numbers, $0$ at irrational numbers, and let $g(x)$ be equal to $0$ at rational numbers, $1$ at irrational numbers. These two functions have no limits at any point, but the product is simply the zero function.

Answer 2

Take a function $h$ such that $\lim_{x\to a}h(x)$ does not exist. Define $$f(x)=\left\{\begin{array}{ll}h(x)&\text{ if }x<a\\0&\text{ if }x>a\end{array}\right.,$$ $$g(x)=\left\{\begin{array}{ll}0&\text{ if }x<a\\h(x)&\text{ if }x>a\end{array}\right..$$