What's the strategy one may use when facing a limit like this one? I think it's more important to know the possible ways to go than the answer itself. It's a problem that came to my mind again when I was working on a different problem.
$\lim_{x\to\infty} \frac{{(x!)}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-x! \log(x!))}{x^2}$
Any suggestion, hint are very welcome.