Limit of the composistion of two functions when the limit of $f$ does not exist?

Question

The theorem about the limit of composition of two real functions $f$ and $g$ is proved here. But it is required that the two limits (of $f$ and $g$) both exist.

I can't understand how to deal with the case in which the limit of $f$ does not exists. In particular I would like to know if the following is correct.

Consider $f(x)$ and $g(x)$ (real functions).

If I find out that $\lim_{x \to x_0} f(x)$ does not exists, can I conclude that $\lim_{x\to x_0 }g(f(x))$ does not exist?

Under what conditions is this correct?

Answer 1

No we can't, let consider as $x \to \infty$

• $f(x)=\sin x$
• $g(x)=1$

but for $g(x)=x$ of course the implication holds.

More in general, I think we need to consider case by case upon the specific functions we are considering. I'm not aware about and I can't exclude the existence of some strange pathological cases.