Example to $\lim f(x)g(x)$ may not exist

Question

Let $A\subset\mathbb{R}$, $c$ a cluster point of $A$ and $f,g:A\rightarrow \mathbb{R}$. Suppose that $f$ is bounded on some neighbourhood of $c$ show by example that if $\lim_{x\rightarrow c}g(x)$ exists, then $\lim_{x\rightarrow c}g(x)f(x)$ may not exist.

By hypothesis, $\lim_{x\rightarrow c}f(x)$. I had some ideas:

• $\lim_{x\rightarrow c}1/x$ does not exist
• define $g(x)=1$, and $f(x)$ by part, then $\lim fg$ does not exist

But I could not find $fg=1/x$ and both $\lim f$, $\lim g$ exist. And I don't think we can multiply functions defined by part...

Is there a classic example of that in Analysis?

Answer 1

Consider $$f(x)=\begin{cases}\sin\frac1x & \text{if }x\ne 0\\5 &\text{otherwise}.\end{cases}$$ and $g$ any non-$0$ constant function with $c=0$.