If $x$ is an integer, we can use Sterling's formula
$$x! \approx. \sqrt{2\pi x} \left(\frac{x}{e}\right)^x$$
Thus,
$$((x+1)!)^{1/(x+1)}\approx. (\sqrt{2\pi (x+1)})^{1/(x+1)}\left(\frac{x+1}{e}\right)$$
and
$$(x!)^{1/x}\approx. (\sqrt{2\pi x})^{1/x}\left(\frac{x}{e}\right)$$
The term $\sqrt{2\pi x}^{1/x}$ and $\sqrt{2\pi (x+1)}^{1/(x+1)}$ can easily be shown to approach $1$ as $x \to \infty$. Thus,
$$((x+1)!)^{1/(x+1)}-(x!)^{1/x}\approx. \frac{x+1}{e}-\frac{x}{e} = 1/e$$
as $x\to \infty$
Aside, we do not need to assume that $x$ is an integer. Indeed, the Gamma function provides an extension of the factorial for complex values. Using the large argument asymptotic expansion of the Gamma function with positive real arguments, we have
$$x!=\Gamma(x+1)=\sqrt{2\pi x}\left(\frac{x}{e}\right)^x \left(1+O\left(\frac{1}{x}\right)\right)$$
Analysis proceeds identically and the result of the post is unaffected (i.e., the limit of the function of this post is $1/e$).