An example of two functions whose limits don't exist at a but the limit of their product does Show by example that the $\lim \limits_{x \to a} [f(x) \cdot g(x)]$ may exist even though neither $\lim \limits_{x \to a} [f(x)]$ nor $\lim \limits_{x \to a} [g(x)]$ exist.
 A: There is an example from the old book for this question. Take for example
$$f(x) =
\begin{cases}
0  & \text{if $x\in\mathbb{Q}$} \\
1 & \text{if $x\notin\mathbb{Q}$}
\end{cases} \quad \text{and} \quad g(x) =
\begin{cases}
1  & \text{if $x\in\mathbb{Q}$} \\
0 & \text{if $x\notin\mathbb{Q}$}
\end{cases}$$
and note that $\displaystyle \lim_{x\to a}f(x)$ and $\displaystyle\lim_{x\to a}g(x)$ does not exist but since that $f(x)g(x)$ is constant function, so the $\displaystyle\lim_{x\to a}f(x)g(x)$ there exists.
A: Consider the limit at $0$ for the functions:
$$f(x) = \chi_{\mathbb R_{\ge 0}} = \begin{cases} 1 &\text{ for } x \ge 0\\0 &\text{ for } x < 0\end{cases}, \quad g(x) = 1-f(x)$$
then both $\lim_{x \to 0} f(x)$ and $\lim_{x \to 0} g(x)$ do not exist.
However $f(x)g(x)\equiv 0$ for all $x$, so $\lim_{x \to 0} f(x)g(x) = 0$.
Alternatively, if $a = \pm \infty$, the limits of $\tan x$ and $\cot x$ do not exist at $a$.
However $\tan x \cot x \equiv 1$ almost everywhere, so the limit of their product is $1$.
A: A very simple example is $f(x)=\sin(\frac{1}{x-a})$ and $g(x)=\frac{1}{f(x)}$: then neither $\lim_{x\to a} f(x)$ nor $\lim_{x\to a} g(x)$ exist, but clearly $\lim_{x\to a} \frac{f(x)}{g(x)}=1$. If you want an example for $x\to (\pm)\infty$, just take $f(x)=\sin(x)$ and again $g(x)=\frac{1}{f(x)}$.
