Tenemos la indeterminación de tipo
oo/oo,
tal que el límite para el numerador es
$$\lim_{n \to \infty} \left(2 n\right)! = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} 2^{n} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(2^{- n} \left(2 n\right)!\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(2 n\right)!}{\frac{d}{d n} 2^{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{2^{1 - n} \Gamma\left(2 n + 1\right)}{\log{\left(2 \right)}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,2 n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{2 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \Gamma\left(2 n + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{2 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \Gamma\left(2 n + 1\right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 n + 1 \right)}\right) \left(\frac{2 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \Gamma\left(2 n + 1\right)\right)^{2}}{- \frac{4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + 4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} - \frac{4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \log{\left(2 \right)} \Gamma\left(2 n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 n + 1 \right)}\right) \left(\frac{2 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \Gamma\left(2 n + 1\right)\right)^{2}}{\frac{d}{d n} \left(- \frac{4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + 4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} - \frac{4 \cdot 2^{1 - n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} - 2^{1 - n} \log{\left(2 \right)} \Gamma\left(2 n + 1\right)\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{5}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{32 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} + \frac{192 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}{\left(1,2 n + 1 \right)}} + \frac{48 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{32 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} - \frac{192 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}^{2}} - 96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} - \frac{96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}} - \frac{8 \cdot 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{8 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} + \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} + 16 \cdot 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} + \frac{16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}{\left(1,2 n + 1 \right)}} - 16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{- \frac{16 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + 24 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} - \frac{48 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} - 12 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} + 24 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)} - \frac{16 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}} + 2 \cdot 2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(2 n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{5}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{32 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{4}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} + \frac{192 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}^{2} \operatorname{polygamma}{\left(1,2 n + 1 \right)}} + \frac{48 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{32 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{64 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} - \frac{192 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}^{2}} - 96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} - \frac{96 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}} - \frac{8 \cdot 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} + \frac{8 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(3,2 n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 n + 1 \right)}} - \frac{16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 n + 1 \right)}} + \frac{128 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} + 16 \cdot 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} + \frac{16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\operatorname{polygamma}{\left(1,2 n + 1 \right)}} - 16 \cdot 2^{- 2 n} \Gamma^{2}\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{- \frac{16 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{3}{\left(0,2 n + 1 \right)}}{\log{\left(2 \right)}} + 24 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}^{2}{\left(0,2 n + 1 \right)} - \frac{48 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} \operatorname{polygamma}{\left(1,2 n + 1 \right)}}{\log{\left(2 \right)}} - 12 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(0,2 n + 1 \right)} + 24 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(1,2 n + 1 \right)} - \frac{16 \cdot 2^{- n} \Gamma\left(2 n + 1\right) \operatorname{polygamma}{\left(2,2 n + 1 \right)}}{\log{\left(2 \right)}} + 2 \cdot 2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(2 n + 1\right)}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)