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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.

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    @Potato: $\lim_{n\to\infty} \frac {(n!)^\frac{1}{n}}{n}$2012-08-05

2 Answers 2

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Note that $ \frac{\left(\Gamma(x+1)^{1/x}\right)'}{(x)'}= \lim_{x\to\infty} \frac{{(\Gamma(x))}^{\frac{1}{x}-1} (x\Gamma(x+1) \psi^{(0)}(x+1)-\Gamma(x) \log(\Gamma(x)))}{x^2} $ So recalling L'Hopital's rule we see that it is enough to find $ \lim\limits_{x\to\infty}\frac{\Gamma(x+1)^{1/x}}{x} $ We know the following asymptotic $ \Gamma(x+1)\sim\left(\frac{x}{e}\right)^x\sqrt{2\pi x}\quad\text{ when }\quad x\to\infty $ then $ \lim\limits_{x\to\infty}\frac{\Gamma(x+1)^{1/x}}{x}= \lim\limits_{x\to\infty}\frac{\frac{x}{e}(2\pi x)^{1/(2x)}}{x}= \lim\limits_{x\to\infty}\frac{1}{e}(2\pi x)^{1/(2x)}=\frac{1}{e} $ As for the another one approach to the limit $ \lim\limits_{x\to\infty}\frac{(x!)^{1/x}}{x} $ see this question.

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    I swear, I solved it without W|A. Expression $\Gamma(x)^{1/x-1}$ reminded me the following formula $(u^v)'=u^{v-1}(u'v+v'\log u)$ The rest was clear.2012-08-05
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The numerator presents the most difficult problem. First, we can see that in this case we can simplify: $\Gamma(x+1) = x!$, so we can pull out a factor $x!$ to get

$(x!)^{1/x}(x \psi^{(0)}(x + 1) - \log(x!))$

(for the numerator.)

This is much more manageable.

Then what I'd do next is to replace the nasty functions ($x!$, $\log(x!)$ and $\psi$) with their elementary asymptotics toward infinity (Stirling's approximation and the digamma's log asymptotic). This should help.