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(12+x)-sqrt(4-x))/(-8+x^2+2*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
- factorial(n)/(- uno + dos *n)
- factorial(n) dividir por ( menos 1 más 2 multiplicar por n)
- factorial(n) dividir por ( menos uno más dos multiplicar por n)
- factorial(n)/(-1+2n)
- factorialn/-1+2n
- factorial(n) dividir por (-1+2*n)
-
Expresiones semejantes
- factorial(n)/(-1-2*n)
- factorial(n)/(1+2*n)
-
Expresiones con funciones
- factorial
- factorial(x)^(1/x)/x
- factorial(n)/(2+n^2)
- factorial(-1+n)/(-factorial(-1+n)+factorial(1+n))
- factorial(-1+x)*exp(x)/factorial(x)
- factorial(x)/(-factorial(n)+factorial(1+x))
Límite de la función factorial(n)/(-1+2*n)
Solución
Ha introducido
[src]
/ n! \ lim |--------| n->oo\-1 + 2*n/
$$\lim_{n \to \infty}\left(\frac{n!}{2 n - 1}\right)$$
Limit(factorial(n)/(-1 + 2*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
$$\lim_{n \to \infty}\left(2 n - 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{n!}{2 n - 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} \left(2 n - 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right)}{2}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2}\right)}{\frac{d}{d n} \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{\operatorname{polygamma}{\left(2,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(2,n + 1 \right)}}}{\frac{d}{d n} \frac{1}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}\right)^{2} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)}}{2} + \frac{7 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}\right) \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}\right)^{2} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}\right)}{\frac{d}{d n} \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)}}{2} + \frac{7 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{9}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{5 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{7}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{27 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)}}{2} + 6 \Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + 5 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2} + 9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{9}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{5 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{7}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{27 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)}}{2} + 6 \Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + 5 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2} + 9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 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
$$\lim_{n \to \infty}\left(2 n - 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{n!}{2 n - 1}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} \left(2 n - 1\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right)}{2}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2}\right)}{\frac{d}{d n} \operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}{\operatorname{polygamma}{\left(2,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(2,n + 1 \right)}}}{\frac{d}{d n} \frac{1}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}\right)^{2} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)}}{2} + \frac{7 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}\right) \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{2} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2}\right)^{2} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}\right)}{\frac{d}{d n} \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)}}{2} + \frac{7 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{9}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{5 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{7}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{27 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)}}{2} + 6 \Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + 5 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2} + 9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{- \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{9}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{8}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{5 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{7}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{27 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(4,n + 1 \right)}}{4 \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}} + \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)} \operatorname{polygamma}^{2}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}^{3}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 \operatorname{polygamma}{\left(2,n + 1 \right)}} - \frac{9 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{6}{\left(0,n + 1 \right)}}{2} + 6 \Gamma\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} + 5 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}}{2} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2} + 9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)} + 3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
Gráfica
Otros límites con n→0, -oo, +oo, 1
$$\lim_{n \to \infty}\left(\frac{n!}{2 n - 1}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{n!}{2 n - 1}\right) = -1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{n!}{2 n - 1}\right) = -1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{n!}{2 n - 1}\right) = 1$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{n!}{2 n - 1}\right) = 1$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{n!}{2 n - 1}\right) = 0$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{n!}{2 n - 1}\right) = -1$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{n!}{2 n - 1}\right) = -1$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{n!}{2 n - 1}\right) = 1$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{n!}{2 n - 1}\right) = 1$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{n!}{2 n - 1}\right) = 0$$
Más detalles con n→-oo