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Useful characterization that compares powers of the id. func with positive func.

Source: Romanian National Olympiad 2016, grade xi, p. 3

August 25, 2019
real analysis

Problem Statement

Let be a real number a, a, and a function f:R>0⟶R>0. 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
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