Interesting Limit (6)

Relevant wiki: Stirling's Formula

$\begin{aligned} L & = \lim_{n \to \infty} \left[\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!} \right] & \small \color{#3D99F6} \text{By Stirling's formula: }n! \thicksim \sqrt{2\pi n}\left(\frac ne \right)^n \\ & = \lim_{n \to \infty} \left[\sqrt[n+1]{\sqrt{2\pi (n+1)}\left(\frac {n+1}e \right)^{n+1}} - \sqrt[n]{\sqrt{2\pi n}\left(\frac ne \right)^n} \right] \\ & = \lim_{n \to \infty} \left[\left(2\pi (n+1)\right)^{\frac 1{2n+2}}\left(\frac {n+1}e \right) - \left(2\pi n\right)^{\frac 1{2n}}\left(\frac ne \right) \right] \\ & = \lim_{n \to \infty} \frac 1e \left((2\pi)^{\frac 1{2n+2}}(n+1)^{\frac {2n+3}{2n+2}} - (2\pi)^{\frac 1{2n}} n^{\frac {2n+1}{2n}} \right) \\ & = \lim_{n \to \infty} \frac 1e \bigg((2\pi)^{\frac 1{2n+2}}(n+1)^{\frac {1+\frac 3{2n}}{1+\frac 1n}} - (2\pi)^{\frac 1{2n}} n^{1+\frac 1{2n}} \bigg) \\ & = \frac 1e (n+1 - n) \\ & = \boxed{\dfrac 1e} \end{aligned}$