Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (8+x)/x^2
-
Límite de (7-x+4*x^2)/(1+3*x)
-
Límite de (3-10*x+3*x^2)/(-3+x^2-2*x)
-
Límite de (3+3*x^2+10*x)/(-3+2*x^2+5*x)
-
Expresiones idénticas
- tres ^(-x)*factorial(x)
- 3 en el grado ( menos x) multiplicar por factorial(x)
- tres en el grado ( menos x) multiplicar por factorial(x)
- 3(-x)*factorial(x)
- 3-x*factorialx
- 3^(-x)factorial(x)
- 3(-x)factorial(x)
- 3-xfactorialx
- 3^-xfactorialx
-
Expresiones semejantes
- 3^(x)*factorial(x)
-
Expresiones con funciones
- factorial
- factorial((1/4)^x)/factorial((1/5)^x)
- factorial(2*n)/(n^2*factorial(-2+2*n))
- factorial(x)/factorial(-x)
- factorial(n)/(factorial(n)+factorial(1+n))
- factorial(1+x)^2*factorial(2*x)/(factorial(x)^2*factorial(2+2*x))
Límite de la función 3^(-x)*factorial(x)
Solución
Ha introducido
[src]
/ -x \ lim \3 *x!/ x->oo
$$\lim_{x \to \infty}\left(3^{- x} x!\right)$$
Limit(3^(-x)*factorial(x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} x! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 3^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(3^{- x} x!\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} 3^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{3^{- x} \Gamma\left(x + 1\right)}{\log{\left(3 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right) \left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right)^{2}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right) \left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} + 4 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}} + 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} + 4 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}} + 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(x + 1\right)}\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
$$\lim_{x \to \infty} x! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 3^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(3^{- x} x!\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} 3^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{3^{- x} \Gamma\left(x + 1\right)}{\log{\left(3 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right) \left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right)^{2}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right) \left(\frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \Gamma\left(x + 1\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} + 4 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}} + 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,x + 1 \right)}} + \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{4 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 12 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{6 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} + 4 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 2 \cdot 3^{- 2 x} \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{- \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)}}{\log{\left(3 \right)}} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - \frac{3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)}}{\log{\left(3 \right)}} - 3 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + 3 \cdot 3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{3^{- x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)}}{\log{\left(3 \right)}} + 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(x + 1\right)}\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 x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(3^{- x} x!\right) = \infty$$
$$\lim_{x \to 0^-}\left(3^{- x} x!\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(3^{- x} x!\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(3^{- x} x!\right) = \frac{1}{3}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(3^{- x} x!\right) = \frac{1}{3}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(3^{- x} x!\right) = \infty \left(-\infty\right)!$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(3^{- x} x!\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(3^{- x} x!\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(3^{- x} x!\right) = \frac{1}{3}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(3^{- x} x!\right) = \frac{1}{3}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(3^{- x} x!\right) = \infty \left(-\infty\right)!$$
Más detalles con x→-oo