Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (1-1/x)^x
-
Límite de (-12+x+x^2)/(sqrt(-2+x)-sqrt(4-x))
-
Límite de (5+x^2)/(-3+x^2)
-
Límite de x^(1/log(-1+e^x))
-
Expresiones idénticas
- factorial(x)/(a+x)
- factorial(x) dividir por (a más x)
- factorialx/a+x
- factorial(x) dividir por (a+x)
-
Expresiones semejantes
- factorial(x)/(a-x)
-
Expresiones con funciones
- factorial
- factorial(x)^(1/x)
- factorial(x)/factorial(1+x)
- factorial(x)/factorial(-1+x)
- factorial(x)/x^2
- factorial(x)/(1+3*x)
Límite de la función factorial(x)/(a+x)
Solución
Ha introducido
[src]
/ x! \ lim |-----| x->oo\a + x/
$$\lim_{x \to \infty}\left(\frac{x!}{a + x}\right)$$
Limit(factorial(x)/(a + x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} x! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(a + x\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x!}{a + x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{\partial}{\partial x} \left(a + x\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 1\right)}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}\right)}{\frac{d}{d x} \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(2,x + 1 \right)}}}{\frac{d}{d x} \frac{1}{- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\left(\Gamma\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} + 7 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}\right)}{\frac{d}{d x} \left(\Gamma\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} + 7 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{9}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{20 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{54 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{9 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{36 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} + 12 \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 10 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 27 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)} + 18 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{9}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{20 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{54 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{9 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{36 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} + 12 \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 10 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 27 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)} + 18 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(1,x + 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
$$\lim_{x \to \infty} x! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(a + x\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x!}{a + x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{\partial}{\partial x} \left(a + x\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \Gamma\left(x + 1\right)}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}\right)}{\frac{d}{d x} \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(2,x + 1 \right)}}}{\frac{d}{d x} \frac{1}{- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\left(\Gamma\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} + 7 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\left(- \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} - 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}\right)^{2} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}\right)}{\frac{d}{d x} \left(\Gamma\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} + 7 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{9}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{20 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{54 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{9 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{36 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} + 12 \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 10 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 27 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)} + 18 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{9}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{8}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{20 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{6 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{54 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} - \frac{9 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(4,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}} + \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} \operatorname{polygamma}^{2}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(2,x + 1 \right)}} - \frac{18 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}{\left(2,x + 1 \right)}} - \frac{36 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} + 12 \Gamma\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + 10 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 27 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)} + 3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)} + 18 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} + 6 \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 5 vez (veces)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x!}{a + x}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x!}{a + x}\right) = \frac{1}{a}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x!}{a + x}\right) = \frac{1}{a}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x!}{a + x}\right) = \frac{1}{a + 1}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x!}{a + x}\right) = \frac{1}{a + 1}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x!}{a + x}\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{x!}{a + x}\right) = \frac{1}{a}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x!}{a + x}\right) = \frac{1}{a}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x!}{a + x}\right) = \frac{1}{a + 1}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x!}{a + x}\right) = \frac{1}{a + 1}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x!}{a + x}\right) = 0$$
Más detalles con x→-oo