Limit inside the exp function

Question

I do not understand the following limit-rule:

$$\lim_{x\to\infty}\exp(f(x))=\exp(\lim_{x\to\infty}f(x))$$

Why is that true?

Answer 1

Since the exponential function $\exp(\cdot) : \mathbb R \to \mathbb R$ is continuous, if the limit $\lim_{x \to \infty} f(x)$ exists, then it can be indeed written that : $$\lim_{x\to\infty}\exp(f(x))=\exp(\lim_{x\to\infty}f(x))$$