Asymptotics of the solution of a differential equation

Let $f$ be twice continuously differentiable function defined on $\mathbb{R}$, such that $f(x)f''(x)=1$ for all $x\ge 0$, $f(0)=1$ and $f'(0)=0$.

Find $$\lim_{x\to\infty} \frac{f(x)}{x\sqrt{\ln(x)}}.$$

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