Is there a function f with $ f (x) $ divergent for large x and slower than logarithmic growth rates?

Question

Problem: Is there a function $f:[0,+\infty]\to \mathbb{R}$ with the following conditions:

1. $f(x)\ge 0$ for $x>C$ where $C$ is constant.
2. $\lim_{x \to \infty} f(x)=+\infty$
3. $\lim_{x\to +\infty} \frac{f(x)}{\ln x} = 0$ or $f'(x) < x^{-1}$

My attempt: take $f(x) =\ln \ln x$ for $x> e$ and conditions 1-3 are satisfied. I am interested in some "better" $f$ in the sense of not using a concatenation of logarithms or arithmetic modifications. For example $f(x)=\ln \ln \ln x$.

Answer 1

Hint

Try $$f(x)=\sqrt{\tan\left({\pi\over 2}\cdot{\ln x\over 1+\ln x}\right)}$$and prove by L^Hoptital's rule that $$\lim_{u\to \infty}{\sqrt{\tan\left({\pi\over 2}\cdot{u\over 1+u}\right)}\over u}=0$$

Answer 2

You can modify your example a bit so that it is defined on $[0,\infty)$

$$ f(x)=\ln(1+\ln(1+x))\text{ for }x\ge0 $$