Proof of growth of two increasing function

Question

Suppose that we have functions $f(n)$ and $g(n)$ and both of them are increasing and both of them don't converge to the constant number and domain of them is Natural numbers and always they are positive. Also suppose that $$\lim \frac{\log g(n)}{\log f(n)} = 0 \text{ for } n \rightarrow \infty$$

How to prove that: $\lim \frac{g(n)}{f(n)} = 0 \text{ for } n \rightarrow \infty$

Answer 1

We know that $g\left(n\right)$ and $f\left(n\right)$ tend to $+\infty$. Write $$ \log g\left(n\right)=\log f\left(n\right)\varepsilon\left(n\right), $$ where $\varepsilon\left(n\right)\rightarrow0$ . Then \begin{align*} g\left(n\right) & =\exp\left(\log f\left(n\right)\varepsilon\left(n\right)\right)=\exp\left(\log f\left(n\right)+\log f\left(n\right)\varepsilon\left(n\right)-\log f\left(n\right)\right)\\ & =f\left(n\right)\exp\left(\log f\left(n\right)\left(\varepsilon\left(n\right)-1\right)\right). \end{align*} Now $\log f\left(n\right)\left(\varepsilon\left(n\right)-1\right)\rightarrow-\infty$, hence its exponential tends to $0$.

Answer 2

I am not sure it is correct. If $g\left(n\right)$ increases to 1 and $f\left(n\right)$ increases to $2$, then $$ \frac{\log\left(g\left(n\right)\right)}{\log\left(f\left(n\right)\right)}\rightarrow0, $$ but $$ \frac{g\left(n\right)}{f\left(n\right)}\rightarrow\frac{1}{2}. $$