Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (9+x^2-6*x)/(x^2-3*x)
-
Límite de (-2+sqrt(-2+x))/(-6+x)
-
Límite de (sqrt(6+x^2-2*x)-sqrt(-6+x^2+2*x))/(3+x^2-4*x)
-
Límite de -sin(sqrt(x))+sin(sqrt(1+x))
-
Expresiones idénticas
- cinco ^(-n)*factorial(n)
- 5 en el grado ( menos n) multiplicar por factorial(n)
- cinco en el grado ( menos n) multiplicar por factorial(n)
- 5(-n)*factorial(n)
- 5-n*factorialn
- 5^(-n)factorial(n)
- 5(-n)factorial(n)
- 5-nfactorialn
- 5^-nfactorialn
-
Expresiones semejantes
- 5^(n)*factorial(n)
-
Expresiones con funciones
- factorial
- factorial(x)^(1/x)/x
- factorial(n)/(2+n^2)
- factorial(-1+x)*exp(x)/factorial(x)
- factorial(x)/(-factorial(n)+factorial(1+x))
- factorial(1+x)^2*factorial(2*x)/(factorial(x)^2*factorial(2+2*x))
Límite de la función 5^(-n)*factorial(n)
Solución
Ha introducido
[src]
/ -n \ lim \5 *n!/ n->oo
$$\lim_{n \to \infty}\left(5^{- n} n!\right)$$
Limit(5^(-n)*factorial(n), n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty} n! = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} 5^{n} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(5^{- n} n!\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} 5^{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{5^{- n} \Gamma\left(n + 1\right)}{\log{\left(5 \right)}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right)^{2}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right)^{2}}{\frac{d}{d n} \left(- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right)\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{8 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}} + 5^{- n} \log{\left(5 \right)}^{2} \Gamma\left(n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{8 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}} + 5^{- n} \log{\left(5 \right)}^{2} \Gamma\left(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)
oo/oo,
tal que el límite para el numerador es
$$\lim_{n \to \infty} n! = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} 5^{n} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(5^{- n} n!\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} 5^{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{5^{- n} \Gamma\left(n + 1\right)}{\log{\left(5 \right)}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right)^{2}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \Gamma\left(n + 1\right)\right)^{2}}{\frac{d}{d n} \left(- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right)\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{8 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}} + 5^{- n} \log{\left(5 \right)}^{2} \Gamma\left(n + 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{4 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{6 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{8 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 n} \log{\left(5 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}} - 2 \cdot 5^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{- \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- n} \log{\left(5 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + 3 \cdot 5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)} - \frac{5^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{\log{\left(5 \right)}} + 5^{- n} \log{\left(5 \right)}^{2} \Gamma\left(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)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(5^{- n} n!\right) = \infty$$
$$\lim_{n \to 0^-}\left(5^{- n} n!\right) = 1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(5^{- n} n!\right) = 1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(5^{- n} n!\right) = \frac{1}{5}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(5^{- n} n!\right) = \frac{1}{5}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(5^{- n} n!\right) = \infty \left(-\infty\right)!$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(5^{- n} n!\right) = 1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(5^{- n} n!\right) = 1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(5^{- n} n!\right) = \frac{1}{5}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(5^{- n} n!\right) = \frac{1}{5}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(5^{- n} n!\right) = \infty \left(-\infty\right)!$$
Más detalles con n→-oo