TITLE: Evaluating the factorial-related limit $\lim_{x \to \infty} (x + 1)!^{1 / (x + 1)} - x!^{1/x}$
QUESTION [10 upvotes]: I'm looking for the limit

$$\lim_{x \to \infty} \left[[(x+1)!]^\frac{1}{1+x} - (x!)^\frac{1}{x}\right].$$

I've put the above in a computer program, and evaluated it at very high values of $x$ (at $x = 100\text{ }000$, it is approximately $0.367881$). The value seems to be caving in to $1/e$, which is $0.3678794412\ldots$
That makes sense, as $e$ has an expansion related to factorial. However I'm stuck trying to figure out a proof if there is any. Thanks for any help.

REPLY [3 votes]: From DeMoivre's formula, we have $$x! \sim C \sqrt{x}\left(\dfrac{x}e\right)^x + \mathcal{O}(1/x)$$
This means we have
\begin{align}
((x+1)!)^{1/(x+1)} - (x!)^{1/x} & \sim C^{1/(x+1)} \sqrt{x}^{1/(x+1)} \left(\dfrac{x+1}e \right) - C^{1/x} \sqrt{x}^{1/x} \left(\dfrac{x}e\right) + \mathcal{O}(1/x)\\
& \sim \dfrac{x+1-x}e = \dfrac1e
\end{align}