Sr Examen
Otras calculadoras:
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¿Cómo usar?
- Límite de la función:
-
Límite de (-9+x^2)/(3+x)
-
Límite de x^2/(1-cos(6*x))
-
Límite de (-3+sqrt(1+2*x))/(sqrt(-2+x)-sqrt(2))
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Límite de (4+x^2-5*x)/(8+x^2-6*x)
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Expresiones idénticas
- (diez + seis *x)/factorial(x)
- (10 más 6 multiplicar por x) dividir por factorial(x)
- (diez más seis multiplicar por x) dividir por factorial(x)
- (10+6x)/factorial(x)
- 10+6x/factorialx
- (10+6*x) dividir por factorial(x)
-
Expresiones semejantes
- (10-6*x)/factorial(x)
-
Expresiones con funciones
- factorial
- factorial(x)^(-1/x)
- factorial(3+x)^3*factorial(1+x)*factorial(2+x)/factorial(x)
- factorial(x)/x^10000
- factorial(1+x)/factorial(x)
- factorial(factorial(-1+2*x))/sqrt((2*n)^n)
Límite de la función (10+6*x)/factorial(x)
Solución
Ha introducido
[src]
/10 + 6*x\ lim |--------| x->oo\ x! /
$$\lim_{x \to \infty}\left(\frac{6 x + 10}{x!}\right)$$
Limit((10 + 6*x)/factorial(x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty}\left(3 x + 5\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(\frac{x!}{2}\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{6 x + 10}{x!}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{2 \left(3 x + 5\right)}{x!}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x + 5\right)}{\frac{d}{d x} \frac{x!}{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6}{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}{\frac{d}{d x} \frac{\Gamma\left(x + 1\right)}{6}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{6 \operatorname{polygamma}{\left(1,x + 1 \right)}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}}{\frac{d}{d x} \left(- \frac{\Gamma\left(x + 1\right)}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 \operatorname{polygamma}{\left(1,x + 1 \right)}}{\left(- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}}\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}}}}{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right)}{6} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(3,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right)}{6} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(3,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \left(\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(4,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{7 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{36 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{4}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{7}{\left(1,x + 1 \right)}} + \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(4,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{7 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{36 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{4}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{7}{\left(1,x + 1 \right)}} + \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(3 x + 5\right) = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(\frac{x!}{2}\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{6 x + 10}{x!}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to \infty}\left(\frac{2 \left(3 x + 5\right)}{x!}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x + 5\right)}{\frac{d}{d x} \frac{x!}{2}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{6}{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}{\frac{d}{d x} \frac{\Gamma\left(x + 1\right)}{6}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{6 \operatorname{polygamma}{\left(1,x + 1 \right)}}{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{\operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}}{\frac{d}{d x} \left(- \frac{\Gamma\left(x + 1\right)}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{3 \operatorname{polygamma}{\left(1,x + 1 \right)}}{\left(- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}}\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{- \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}}}}{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right)}{6} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(3,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right)}{6} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(3,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \left(\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{108 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(4,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{7 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{36 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{4}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{7}{\left(1,x + 1 \right)}} + \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{6 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{2 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{3 \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(4,x + 1 \right)}}{6 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{4}{\left(1,x + 1 \right)}}}{\frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{7}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{54 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{108 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{6}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{27 \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{7 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{36 \operatorname{polygamma}^{6}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{4}{\left(2,x + 1 \right)}}{18 \operatorname{polygamma}^{7}{\left(1,x + 1 \right)}} + \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{27 \operatorname{polygamma}^{4}{\left(1,x + 1 \right)}} + \frac{5 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{3}{\left(2,x + 1 \right)}}{27 \operatorname{polygamma}^{5}{\left(1,x + 1 \right)}} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{9 \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(\frac{6 x + 10}{x!}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{6 x + 10}{x!}\right) = 10$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{6 x + 10}{x!}\right) = 10$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{6 x + 10}{x!}\right) = 16$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{6 x + 10}{x!}\right) = 16$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{6 x + 10}{x!}\right) = - \frac{\infty}{\left(-\infty\right)!}$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{6 x + 10}{x!}\right) = 10$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{6 x + 10}{x!}\right) = 10$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{6 x + 10}{x!}\right) = 16$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{6 x + 10}{x!}\right) = 16$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{6 x + 10}{x!}\right) = - \frac{\infty}{\left(-\infty\right)!}$$
Más detalles con x→-oo