Limits ( 2 )

$\begin{aligned} L & = \lim_{n \to \infty} \left(\frac {(n+1)^2}{\sqrt[n+1]{(n+1)!}} - \frac {n^2}{\sqrt[n]{n!}} \right) & \small \blue{\text{By Stirling's formula: }n! \sim \sqrt{2n\pi}\left(\frac ne\right)^n} \\ & = \lim_{n \to \infty} \left(\frac {(n+1)^2}{(2(n+1)\pi)^{\frac 1{2(n+1)}} \cdot \frac {n+1}e} - \frac {n^2}{(2n\pi)^{\frac 1{2n}} \cdot \frac ne} \right) \\ & = \lim_{n \to \infty} \left(\frac {(n+1)e}{(2(n+1)\pi)^{\frac 1{2(n+1)}}} - \frac {ne}{(2n\pi)^{\frac 1{2n}}} \right) & \small \blue{\text{Note that }\lim_{n \to \infty} (2\pi)^{\frac 1{2(n+1)}} = \lim_{n \to \infty} {2\pi}^{\frac 1{2n}} = 1} \\ & = \lim_{n \to \infty} \left(e\blue{n\left((n+1)^{-\frac 1{2(n+1)}} - n^{-\frac 1{2n}} \right)} + e\red{(n+1)^{-\frac 1{2(n+1)}}}\right) & \small \blue{ \text{By theorem 1 (see note)}} \\ & = e \blue{\lim_{n \to \infty} n(n+1-n)\frac {d\ n^{-\frac 1{2n}}}{dn}} + \exp \left(1-\red{\lim_{n \to \infty} \frac {\ln (n+1)}{2(n+1)}} \right) & \small \red{\text{An }\infty/\infty \text{ case, L'Hôpital's rule applies.}} \\ & = e \blue{\lim_{n \to \infty} n^{1-\frac 1{2n}}\left(-\frac 1{2n^2}+\frac {\ln n}{2n^2}\right)} + \exp \left(1-\red{\lim_{n \to \infty} \frac {\frac 1{n+1}}2} \right) & \small \red{\text{Differentiate up and down w.r.t. }n} \\ & = \blue 0 + e^{1-\red 0} = \boxed e \end{aligned}$

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References:

• Theorem 1: $\displaystyle \lim_{n \to \infty} n(f(a_{n+1} - f(a_n)) = \lim_{n \to \infty} n(a_{n+1} - a_n) f'(a_n)$ . Source: New Methods For Calculations of Some Limits
• L'Hôpital's rule