Under what conditions does $\lim_{x\to\infty} f(g(x))=f \left(\lim_{x\to\infty} g(x) \right)$?

Question

I know that if $a\in\mathbb{R}$ and $f$ is continuous at $a$, then $$\lim_{x\to a} f(g(x))=f\left(\lim_{x\to a} g(x)\right).$$

However, this does not easily translate to the case where $a=\infty$. Can anyone help me here?

Answer 1

Suppose that $\lim_{x\to\infty}g(x)=a(\in\mathbb{R})$ and that $f$ is continuous at $a$. Then$$\lim_{x\to\infty}f\bigl(g(x)\bigr)=f\left(\lim_{x\to\infty}g(x)\right).$$