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Límite de (9+x^2-6*x)/(x^2-3*x)
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Límite de (sqrt(6+x^2-2*x)-sqrt(-6+x^2+2*x))/(3+x^2-4*x)
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Expresiones idénticas
- dos ^(-n)*factorial(n)/ dos
- 2 en el grado ( menos n) multiplicar por factorial(n) dividir por 2
- dos en el grado ( menos n) multiplicar por factorial(n) dividir por dos
- 2(-n)*factorial(n)/2
- 2-n*factorialn/2
- 2^(-n)factorial(n)/2
- 2(-n)factorial(n)/2
- 2-nfactorialn/2
- 2^-nfactorialn/2
- 2^(-n)*factorial(n) dividir por 2
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Expresiones semejantes
- 2^(n)*factorial(n)/2
-
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 2^(-n)*factorial(n)/2
Solución
Ha introducido
[src]
/ -n \
|2 *n!|
lim |------|
n->oo\ 2 /
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right)$$
Limit((2^(-n)*factorial(n))/2, n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty}\left(\frac{n!}{2}\right) = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{n!}{2}}{\frac{d}{d n} 2^{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{2^{- n} \Gamma\left(n + 1\right)}{2 \log{\left(2 \right)}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right)^{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right)}{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right)^{2}}{\frac{d}{d n} \left(- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right)}{2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}} + \frac{2^{- 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(2 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 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(2 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}^{2}} - 3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(2 \right)}} + 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2} - \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{3 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(n + 1\right)}{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}} + \frac{2^{- 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(2 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 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(2 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}^{2}} - 3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(2 \right)}} + 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2} - \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{3 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(n + 1\right)}{2}}\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
$$\lim_{n \to \infty}\left(\frac{n!}{2}\right) = \infty$$
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.
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{n!}{2}}{\frac{d}{d n} 2^{n}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{2^{- n} \Gamma\left(n + 1\right)}{2 \log{\left(2 \right)}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right)^{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right)}{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right) \left(\frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \Gamma\left(n + 1\right)}{2}\right)^{2}}{\frac{d}{d n} \left(- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right)}{2}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}} + \frac{2^{- 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(2 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 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(2 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}^{2}} - 3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(2 \right)}} + 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2} - \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{3 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(n + 1\right)}{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}^{2}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{\log{\left(2 \right)}} + \frac{2^{- 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(2 \right)}^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 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(2 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}^{2}} - 3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} - \frac{3 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{2 \cdot 2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{\log{\left(2 \right)}} + 2^{- 2 n} \log{\left(2 \right)} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} + \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{2^{- 2 n} \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{- \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2} - \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 \log{\left(2 \right)}} - \frac{3 \cdot 2^{- n} \log{\left(2 \right)} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2} + \frac{3 \cdot 2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} - \frac{2^{- n} \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 \log{\left(2 \right)}} + \frac{2^{- n} \log{\left(2 \right)}^{2} \Gamma\left(n + 1\right)}{2}}\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
$$\lim_{n \to \infty}\left(\frac{2^{- n} n!}{2}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{4}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{4}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{2^{- n} n!}{2}\right) = \infty \left(-\infty\right)!$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{4}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{2^{- n} n!}{2}\right) = \frac{1}{4}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{2^{- n} n!}{2}\right) = \infty \left(-\infty\right)!$$
Más detalles con n→-oo