I came up with the following exercise :

Let $f$ be a $\mathscr{C}^{1}$ and positive function in $\left[1,+\infty\right[$ such as $f\left(1\right)=1$.

We suppose that
$$
\frac{xf'\left(x\right)}{f\left(x\right)}\underset{x \rightarrow +\infty}{\rightarrow} \alpha
$$
where $\alpha \in \mathbb{R}$.

$(i)$ Find the limit of $\displaystyle \text{ }\frac{\ln\left(f\left(x\right)\right)}{\ln\left(x\right)}$ when $x \rightarrow +\infty$

$(ii)$ Deduce that if $\alpha<-1$, $f$ is integrable in $\left[1,+\infty\right[$.

*What I have done :*

I used that
$$
\int\_{1}^{x}t\frac{f'\left(t\right)}{f\left(t\right)}\text{d}t=\left[t\ln\left(f\left(t\right)\right)\right]^{x}\_{1}-\int\_{1}^{x}\ln\left(f\left(t\right)\right)\text{d}t=x\ln\left(f\left(x\right)\right)-\int\_{1}^{x}\ln\left(f\left(t\right)\right)\text{d}t
$$
But well doesn't work well ... Do you guys have suggestions ?