Límite de la función factorial(x)/x^23

Tenemos la indeterminación de tipo

oo/oo,

tal que el límite para el numerador es
$$\lim_{x \to \infty} x! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} x^{23} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x!}{x^{23}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} x^{23}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{22}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{\Gamma\left(x + 1\right)}{23 x^{22}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{22 \Gamma\left(x + 1\right)}{23 x^{23}}\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{22 \Gamma\left(x + 1\right)}{23 x^{23}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{22 \Gamma\left(x + 1\right)}{23 x^{23}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{22}} + \frac{44 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{23}} - \frac{22 \Gamma\left(x + 1\right)}{x^{24}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{22 \Gamma\left(x + 1\right)}{23 x^{23}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{\frac{d}{d x} \left(- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{22}} + \frac{44 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{23}} - \frac{22 \Gamma\left(x + 1\right)}{x^{24}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{529 x^{44}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{44}} - \frac{132 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{264 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{529 x^{45}} - \frac{44 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{88 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{132 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{176 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{45}} + \frac{2948 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{5896 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{529 x^{46}} + \frac{484 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{46}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{23 x^{47} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{1936 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{47}}}{- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{22}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{23 x^{22}} + \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{23 x^{23}} + \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{23}} - \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{24}} + \frac{528 \Gamma\left(x + 1\right)}{x^{25}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{529 x^{44}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{44} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{44}} - \frac{132 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{264 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{529 x^{45}} - \frac{44 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{88 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{132 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{45} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{176 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{45}} + \frac{2948 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{5896 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{529 x^{46}} + \frac{484 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{529 x^{46} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{529 x^{46}} - \frac{968 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{23 x^{47} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{1936 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{23 x^{47}}}{- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{23 x^{22}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{22}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{23 x^{22}} + \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{23 x^{23}} + \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{23 x^{23}} - \frac{66 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{24}} + \frac{528 \Gamma\left(x + 1\right)}{x^{25}}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)