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- Límite de la función:
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Límite de (9+x^2-6*x)/(x^2-3*x)
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Límite de (-2+sqrt(-2+x))/(-6+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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Límite de -sin(sqrt(x))+sin(sqrt(1+x))
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Expresiones idénticas
- veintisiete *factorial(n)/n^ tres
- 27 multiplicar por factorial(n) dividir por n al cubo
- veintisiete multiplicar por factorial(n) dividir por n en el grado tres
- 27*factorial(n)/n3
- 27*factorialn/n3
- 27*factorial(n)/n³
- 27*factorial(n)/n en el grado 3
- 27factorial(n)/n^3
- 27factorial(n)/n3
- 27factorialn/n3
- 27factorialn/n^3
- 27*factorial(n) dividir por n^3
-
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 27*factorial(n)/n^3
Solución
Ha introducido
[src]
/27*n!\
lim |-----|
n->oo| 3 |
\ n /
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right)$$
Limit((27*factorial(n))/n^3, n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty}\left(27 n!\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} n^{3} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 27 n!}{\frac{d}{d n} n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{9 \Gamma\left(n + 1\right)}{n^{2}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} + \frac{36 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} - \frac{54 \Gamma\left(n + 1\right)}{n^{4}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{\frac{d}{d n} \left(- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} + \frac{36 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} - \frac{54 \Gamma\left(n + 1\right)}{n^{4}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{4}} + \frac{81 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{486 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{4}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{5}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{1296 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{5}} + \frac{2268 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4536 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{6}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{6}} - \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{7} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3888 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{7}}}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{2}} - \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{3}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} - \frac{162 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{4}} + \frac{216 \Gamma\left(n + 1\right)}{n^{5}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{4}} + \frac{81 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{486 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{4}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{5}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{1296 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{5}} + \frac{2268 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4536 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{6}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{6}} - \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{7} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3888 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{7}}}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{2}} - \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{3}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} - \frac{162 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{4}} + \frac{216 \Gamma\left(n + 1\right)}{n^{5}}}\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
$$\lim_{n \to \infty}\left(27 n!\right) = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty} n^{3} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} 27 n!}{\frac{d}{d n} n^{3}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{9 \Gamma\left(n + 1\right)}{n^{2}}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} + \frac{36 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} - \frac{54 \Gamma\left(n + 1\right)}{n^{4}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{18 \Gamma\left(n + 1\right)}{n^{3}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{\frac{d}{d n} \left(- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} + \frac{36 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} - \frac{54 \Gamma\left(n + 1\right)}{n^{4}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{4}} + \frac{81 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{486 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{4}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{5}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{1296 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{5}} + \frac{2268 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4536 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{6}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{6}} - \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{7} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3888 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{7}}}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{2}} - \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{3}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} - \frac{162 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{4}} + \frac{216 \Gamma\left(n + 1\right)}{n^{5}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{4}} + \frac{81 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{162 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{486 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{4}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{5}} - \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{972 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{5} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{1296 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{5}} + \frac{2268 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{4536 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{6}} + \frac{324 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{6} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{648 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{6}} - \frac{1944 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{7} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3888 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{7}}}{- \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{2}} - \frac{27 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{2}} - \frac{9 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{3}} + \frac{54 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} - \frac{162 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{4}} + \frac{216 \Gamma\left(n + 1\right)}{n^{5}}}\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
$$\lim_{n \to \infty}\left(\frac{27 n!}{n^{3}}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{27 n!}{n^{3}}\right) = -\infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{27 n!}{n^{3}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{27 n!}{n^{3}}\right) = 27$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{27 n!}{n^{3}}\right) = 27$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{27 n!}{n^{3}}\right) = 0$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{27 n!}{n^{3}}\right) = -\infty$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{27 n!}{n^{3}}\right) = \infty$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{27 n!}{n^{3}}\right) = 27$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{27 n!}{n^{3}}\right) = 27$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{27 n!}{n^{3}}\right) = 0$$
Más detalles con n→-oo