Límite de la función factorial(1+x)/(1+x)

Solución

Ha introducido [src]

     /(1 + x)!\
 lim |--------|
x->oo\ 1 + x  /

$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)!}{x + 1}\right)$$

Limit(factorial(1 + x)/(1 + x), x, oo, dir='-')

Método de l'Hopital

Tenemos la indeterminación de tipo

oo/oo,

tal que el límite para el numerador es
$$\lim_{x \to \infty} \left(x + 1\right)! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(x + 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)!}{x + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + 1\right)!}{\frac{d}{d x} \left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 2\right)}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 2 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}\right)}{\frac{d}{d x} \operatorname{polygamma}{\left(1,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} - 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\operatorname{polygamma}{\left(2,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(2,x + 2 \right)}}}{\frac{d}{d x} \frac{1}{- \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} - 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(- \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} - 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\left(\Gamma\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} + 7 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} + 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 6 \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}\right) \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\left(- \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} - 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}\right)}{\frac{d}{d x} \left(\Gamma\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} + 7 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} + 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 6 \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{9}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{\Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{8}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{2 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{8}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{20 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{7}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{6 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{12 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{6 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}{\left(2,x + 2 \right)}} - \frac{54 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{9 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{18 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{18 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}{\left(2,x + 2 \right)}} - \frac{36 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}}{\Gamma\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} + 12 \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} + 10 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 27 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} + 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)} + 18 \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 6 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{9}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{\Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{8}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{2 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{8}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{20 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{7}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{6 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{12 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{6 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}{\left(2,x + 2 \right)}} - \frac{54 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} - \frac{9 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(4,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}} + \frac{18 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} \operatorname{polygamma}^{2}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 2 \right)}} - \frac{18 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}{\left(2,x + 2 \right)}} - \frac{36 \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}}{\Gamma\left(x + 2\right) \operatorname{polygamma}^{6}{\left(0,x + 2 \right)} + 12 \Gamma\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} + 10 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 27 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)} + 3 \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)} + 18 \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)} + 6 \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)

Respuesta rápida [src]

oo

$$\infty$$

Abrir y simplificar

Otros límites con x→0, -oo, +oo, 1

$$\lim_{x \to \infty}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = 1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = 1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(x + 1\right)!}{x + 1}\right) = 0$$
Más detalles con x→-oo

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Límite de la función factorial(1+x)/(1+x)

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