RO-NMO 2016 Problem 113

Let be a real number $a,$ and a function $f:\mathbb{R}_{>0 }\longrightarrow\mathbb{R}_{>0 } .$ Show that the following relations are equivalent. $\text{(i)} \varepsilon\in\mathbb{R}_{>0 } \implies\left( \lim_{x\to\infty } \frac{f(x)}{x^{a+\varepsilon }} =0\wedge \lim_{x\to\infty } \frac{f(x)}{x^{a-\varepsilon }} =\infty \right)$ $\text{(ii)} \lim_{x\to\infty } \frac{\ln f(x)}{\ln x } =a$