f(x)g(x) is not an indeterminate form?

Question

Why is it that when $\lim_{x \to \infty} g(x) = \infty$ and $\lim_{x \to \infty} f(x) = 1$ , $g(x)f(x)$ is not an indeterminate form?

Answer 1

Rigorous proof :

Let $C$ be an arbitary positive real number. For sufficient large $x$ we have $$f(x)\ge 0.99$$ and $$g(x)\ge C$$

Hence $$f(x)g(x)\ge 0.99\cdot C$$

Since $C$ was arbitary, the limit must be $\infty$