On the existence of $ \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} $ under certain constraints.

Question

I am considering a continuously differentiable real-valued function $ f: (0,1) \to (0,\infty) $ such that:

1. $ f $ is decreasing on $ (0,1) $.
2. $ \displaystyle \lim_{x \to 0^{+}} f(x) = \infty $.
3. The map $ x \mapsto x^{2} f'(x) $ is decreasing on $ (0,1) $.
4. $ \displaystyle \lim_{x \to 0^{+}} x^{2} f'(x) = 0 $.

I am wondering if, under these constraints, the limit $ \displaystyle \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} $ exists.

Thanks a lot to anybody who has any thought/counterexample, or spent time reading this question!