Question 63
Let $a \in \mathbb{R}$ and $f : (0, \infty) \to (0, \infty)$. Prove that the following two statements are equivalent:
(i) $\lim_{x \to \infty} \frac{f(x)}{x^{a+\varepsilon}} = 0$ and $\lim_{x \to \infty} \frac{f(x)}{x^{a-\varepsilon}} = \infty$, for all $\varepsilon \gt 0$;
(ii) $\lim_{x \to \infty} \frac{\ln f(x)}{\ln x} = a$.
(i) $\lim_{x \to \infty} \frac{f(x)}{x^{a+\varepsilon}} = 0$ and $\lim_{x \to \infty} \frac{f(x)}{x^{a-\varepsilon}} = \infty$, for all $\varepsilon \gt 0$;
(ii) $\lim_{x \to \infty} \frac{\ln f(x)}{\ln x} = a$.