Connection between limit of ratio of logarithms of functions and functions

Question

Suppose $f(x)$ and $g(x)$ both are positive increasing functions and both approach to $\infty$ as $x \to \infty$. We know $lim_{x \to \infty} log (f(x))/log(g(x)) =0$. Can we say $lim_{x \to \infty} f(x)/g(x) =0$?

logarithms

Answer 1

Original Question:

Try $f(x)=1-1/x^{2}$ and $g(x)=1-1/x$, this is a counterexample.

EDIT:

For $0<\epsilon<1$, then \begin{align*} \dfrac{\log f(x)}{\log g(x)}<\epsilon \end{align*} for large $x$, it entails \begin{align*} \dfrac{f(x)}{g(x)}<\dfrac{1}{g(x)}\cdot g(x)^{\epsilon}=\dfrac{1}{g(x)^{1-\epsilon}}\rightarrow 0 \end{align*} since $g(x)\rightarrow\infty$.