Tenemos la indeterminación de tipo
oo/oo,
tal que el límite para el numerador es
$$\lim_{n \to \infty} n! = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} \sqrt{n} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{n!}{\sqrt{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} \sqrt{n}}\right)$$
=
$$\lim_{n \to \infty}\left(2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 2 \sqrt{n} \Gamma\left(n + 1\right)}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{\Gamma\left(n + 1\right)}{\sqrt{n}}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{\Gamma\left(n + 1\right)}{\sqrt{n}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{\Gamma\left(n + 1\right)}{\sqrt{n}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{- 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{2 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\sqrt{n}} + \frac{\Gamma\left(n + 1\right)}{2 n^{\frac{3}{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{\Gamma\left(n + 1\right)}{\sqrt{n}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{\frac{d}{d n} \left(- 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{2 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\sqrt{n}} + \frac{\Gamma\left(n + 1\right)}{2 n^{\frac{3}{2}}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{8 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 16 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} + \frac{4 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{8 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{16 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 24 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + \frac{12 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 24 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} + \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{8 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 16 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}}}{- 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} - 6 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} - 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\sqrt{n}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\sqrt{n}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n^{\frac{3}{2}}} - \frac{3 \Gamma\left(n + 1\right)}{4 n^{\frac{5}{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{8 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 16 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} + \frac{4 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{8 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{16 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 24 n \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + \frac{12 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 24 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} + \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{8 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 16 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}}}{- 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} - 6 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} - 2 \sqrt{n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\sqrt{n}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\sqrt{n}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n^{\frac{3}{2}}} - \frac{3 \Gamma\left(n + 1\right)}{4 n^{\frac{5}{2}}}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)