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
- factorial(n)/(dos +n^ dos)
- factorial(n) dividir por (2 más n al cuadrado )
- factorial(n) dividir por (dos más n en el grado dos)
- factorial(n)/(2+n2)
- factorialn/2+n2
- factorial(n)/(2+n²)
- factorial(n)/(2+n en el grado 2)
- factorialn/2+n^2
- factorial(n) dividir por (2+n^2)
-
Expresiones semejantes
- factorial(n)/(2-n^2)
-
Expresiones con funciones
- factorial
- factorial(x)^(1/x)/x
- 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))
- factorial(x)/factorial(-x)
Límite de la función factorial(n)/(2+n^2)
Solución
Ha introducido
[src]
/ n! \
lim |------|
n->oo| 2|
\2 + n /
$$\lim_{n \to \infty}\left(\frac{n!}{n^{2} + 2}\right)$$
Limit(factorial(n)/(2 + n^2), n, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{n \to \infty} n! = \infty$$
y el límite para el denominador es
$$\lim_{n \to \infty}\left(n^{2} + 2\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{n!}{n^{2} + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} \left(n^{2} + 2\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right)}{2 n}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{\Gamma\left(n + 1\right)}{n^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{\frac{d}{d n} \left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{\Gamma\left(n + 1\right)}{n^{3}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{3}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{3} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} + \frac{2 \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}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{4}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{4}} - \frac{\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}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{5}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 n} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n^{2}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} + \frac{3 \Gamma\left(n + 1\right)}{n^{4}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{3}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{3} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} + \frac{2 \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}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{4}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{4}} - \frac{\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}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{5}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 n} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n^{2}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} + \frac{3 \Gamma\left(n + 1\right)}{n^{4}}}\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
$$\lim_{n \to \infty}\left(n^{2} + 2\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{n \to \infty}\left(\frac{n!}{n^{2} + 2}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} n!}{\frac{d}{d n} \left(n^{2} + 2\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \frac{\Gamma\left(n + 1\right)}{2 n}}{\frac{d}{d n} \frac{1}{\operatorname{polygamma}{\left(0,n + 1 \right)}}}\right)$$
=
$$\lim_{n \to \infty}\left(- \frac{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{\operatorname{polygamma}{\left(1,n + 1 \right)}}\right)}{\frac{d}{d n} \frac{1}{\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{\Gamma\left(n + 1\right)}{n^{3}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{d}{d n} \left(\frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right)}{2 n^{2}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,n + 1 \right)}\right)}{\frac{d}{d n} \left(- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} + \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{2}} - \frac{\Gamma\left(n + 1\right)}{n^{3}}\right)}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{3}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{3} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} + \frac{2 \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}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{4}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{4}} - \frac{\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}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{5}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 n} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n^{2}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} + \frac{3 \Gamma\left(n + 1\right)}{n^{4}}}\right)$$
=
$$\lim_{n \to \infty}\left(\frac{\frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{5}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)}}{n^{2}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{2} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{2} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{n^{2} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{4}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{n^{3}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{n^{3} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} - \frac{3 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{3} \operatorname{polygamma}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{n^{3}} + \frac{2 \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}^{2}{\left(1,n + 1 \right)}} - \frac{4 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{n^{4}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(3,n + 1 \right)}}{4 n^{4} \operatorname{polygamma}^{2}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)} \operatorname{polygamma}^{2}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}^{3}{\left(1,n + 1 \right)}} + \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n^{4} \operatorname{polygamma}{\left(1,n + 1 \right)}} - \frac{\Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{4}} - \frac{\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}^{2}{\left(1,n + 1 \right)}} + \frac{2 \Gamma^{2}\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{5}}}{- \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}^{3}{\left(0,n + 1 \right)}}{2 n} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)} \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n} - \frac{\Gamma\left(n + 1\right) \operatorname{polygamma}{\left(2,n + 1 \right)}}{2 n} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}^{2}{\left(0,n + 1 \right)}}{2 n^{2}} + \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(1,n + 1 \right)}}{2 n^{2}} - \frac{3 \Gamma\left(n + 1\right) \operatorname{polygamma}{\left(0,n + 1 \right)}}{n^{3}} + \frac{3 \Gamma\left(n + 1\right)}{n^{4}}}\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
$$\lim_{n \to \infty}\left(\frac{n!}{n^{2} + 2}\right) = \infty$$
$$\lim_{n \to 0^-}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{3}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{3}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{n!}{n^{2} + 2}\right) = 0$$
Más detalles con n→-oo
$$\lim_{n \to 0^-}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la izquierda
$$\lim_{n \to 0^+}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{2}$$
Más detalles con n→0 a la derecha
$$\lim_{n \to 1^-}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{3}$$
Más detalles con n→1 a la izquierda
$$\lim_{n \to 1^+}\left(\frac{n!}{n^{2} + 2}\right) = \frac{1}{3}$$
Más detalles con n→1 a la derecha
$$\lim_{n \to -\infty}\left(\frac{n!}{n^{2} + 2}\right) = 0$$
Más detalles con n→-oo