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Expresiones idénticas
- cinco ^(-x)*factorial(uno +x)
- 5 en el grado ( menos x) multiplicar por factorial(1 más x)
- cinco en el grado ( menos x) multiplicar por factorial(uno más x)
- 5(-x)*factorial(1+x)
- 5-x*factorial1+x
- 5^(-x)factorial(1+x)
- 5(-x)factorial(1+x)
- 5-xfactorial1+x
- 5^-xfactorial1+x
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Expresiones semejantes
- 5^(-x)*factorial(1-x)
- 5^(x)*factorial(1+x)
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Expresiones con funciones
- factorial
- factorial(2+n)
- factorial((1/4)^x)/factorial((1/5)^x)
- factorial(n)+factorial(1+n)
- factorial(3+x)*tan(pi*3^(-1-x))/(factorial(2+x)*tan(pi*3^(-x)))
- factorial(2*n)/(n^2*factorial(-2+2*n))
Límite de la función 5^(-x)*factorial(1+x)
Solución
Ha introducido
[src]
/ -x \ lim \5 *(1 + x)!/ x->oo
$$\lim_{x \to \infty}\left(5^{- x} \left(x + 1\right)!\right)$$
Limit(5^(-x)*factorial(1 + x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} \left(x + 1\right)! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 5^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(5^{- x} \left(x + 1\right)!\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + 1\right)!}{\frac{d}{d x} 5^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{5^{- x} \Gamma\left(x + 2\right)}{\log{\left(5 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 2 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 2 \right)}\right) \left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right)^{2}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 2 \right)}\right) \left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,x + 2 \right)}} + \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{8 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}} - 2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}} + 5^{- x} \log{\left(5 \right)}^{2} \Gamma\left(x + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,x + 2 \right)}} + \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{8 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}} - 2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}} + 5^{- x} \log{\left(5 \right)}^{2} \Gamma\left(x + 2\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
$$\lim_{x \to \infty} \left(x + 1\right)! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 5^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(5^{- x} \left(x + 1\right)!\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x + 1\right)!}{\frac{d}{d x} 5^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{5^{- x} \Gamma\left(x + 2\right)}{\log{\left(5 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 2 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 2 \right)}\right) \left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right)^{2}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 2 \right)}\right) \left(\frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \Gamma\left(x + 2\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 2 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,x + 2 \right)}} + \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{8 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}} - 2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}} + 5^{- x} \log{\left(5 \right)}^{2} \Gamma\left(x + 2\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{5}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{4}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}^{2} \operatorname{polygamma}{\left(1,x + 2 \right)}} + \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{4 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}^{2}} - 12 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{6 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} + \frac{5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(3,x + 2 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 2 \right)}} - \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} \operatorname{polygamma}^{2}{\left(2,x + 2 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 2 \right)}} + \frac{8 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} + 4 \cdot 5^{- 2 x} \log{\left(5 \right)} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + \frac{2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(2,x + 2 \right)}}{\operatorname{polygamma}{\left(1,x + 2 \right)}} - 2 \cdot 5^{- 2 x} \Gamma^{2}\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)}}{- \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{3}{\left(0,x + 2 \right)}}{\log{\left(5 \right)}} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}^{2}{\left(0,x + 2 \right)} - \frac{3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} \operatorname{polygamma}{\left(1,x + 2 \right)}}{\log{\left(5 \right)}} - 3 \cdot 5^{- x} \log{\left(5 \right)} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(0,x + 2 \right)} + 3 \cdot 5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(1,x + 2 \right)} - \frac{5^{- x} \Gamma\left(x + 2\right) \operatorname{polygamma}{\left(2,x + 2 \right)}}{\log{\left(5 \right)}} + 5^{- x} \log{\left(5 \right)}^{2} \Gamma\left(x + 2\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
$$\lim_{x \to \infty}\left(5^{- x} \left(x + 1\right)!\right) = \infty$$
$$\lim_{x \to 0^-}\left(5^{- x} \left(x + 1\right)!\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(5^{- x} \left(x + 1\right)!\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(5^{- x} \left(x + 1\right)!\right) = \frac{2}{5}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(5^{- x} \left(x + 1\right)!\right) = \frac{2}{5}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(5^{- x} \left(x + 1\right)!\right) = \infty \left(-\infty\right)!$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(5^{- x} \left(x + 1\right)!\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(5^{- x} \left(x + 1\right)!\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(5^{- x} \left(x + 1\right)!\right) = \frac{2}{5}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(5^{- x} \left(x + 1\right)!\right) = \frac{2}{5}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(5^{- x} \left(x + 1\right)!\right) = \infty \left(-\infty\right)!$$
Más detalles con x→-oo