Sr Examen
Otras calculadoras:
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- Límite de la función:
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Límite de (1+11/x)^x
-
Límite de (2+x^3-x-2*x^2)/(6+x^3-7*x)
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Límite de (-3+x^2-2*x)/(-15-4*x+3*x^2)
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Límite de (sqrt(5+x)-sqrt(10))/(-15+x^2-2*x)
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Expresiones idénticas
- factorial(dos *x)/(uno + tres ^x)
- factorial(2 multiplicar por x) dividir por (1 más 3 en el grado x)
- factorial(dos multiplicar por x) dividir por (uno más tres en el grado x)
- factorial(2*x)/(1+3x)
- factorial2*x/1+3x
- factorial(2x)/(1+3^x)
- factorial(2x)/(1+3x)
- factorial2x/1+3x
- factorial2x/1+3^x
- factorial(2*x) dividir por (1+3^x)
-
Expresiones semejantes
- factorial(2*x)/(1-3^x)
-
Expresiones con funciones
- factorial
- factorial(2*x)/x^3
- factorial(1+2*x)/(factorial(1+x)*factorial(-1+2*x))
- factorial(1+n)/(factorial(-1+n)+factorial(-2+n))
- factorial(n)^3/factorial(3*n)
- factorial(8*n)/(-3+factorial(n))
Límite de la función factorial(2*x)/(1+3^x)
Solución
Ha introducido
[src]
/(2*x)!\
lim |------|
x->oo| x|
\1 + 3 /
$$\lim_{x \to \infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right)$$
Limit(factorial(2*x)/(1 + 3^x), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} \left(2 x\right)! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(3^{x} + 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(2 x\right)!}{\frac{d}{d x} \left(3^{x} + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)}{\log{\left(3 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,2 x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 x + 1 \right)}\right) \left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right)^{2}}{- \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} - \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 x + 1 \right)}\right) \left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} - \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{5}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} + \frac{48 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} - \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} - \frac{8 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} + 16 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}{\left(1,2 x + 1 \right)}} - 16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{- \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{48 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 12 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)} - \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(2 x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{5}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} + \frac{48 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} - \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} - \frac{8 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} + 16 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}{\left(1,2 x + 1 \right)}} - 16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{- \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{48 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 12 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)} - \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(2 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
$$\lim_{x \to \infty} \left(2 x\right)! = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty}\left(3^{x} + 1\right) = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(2 x\right)!}{\frac{d}{d x} \left(3^{x} + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)}{\log{\left(3 \right)}}}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,2 x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{2 \operatorname{polygamma}{\left(1,2 x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 x + 1 \right)}\right) \left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right)^{2}}{- \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} - \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,2 x + 1 \right)}\right) \left(\frac{4 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \Gamma\left(2 x + 1\right)\right)^{2}}{\frac{d}{d x} \left(- \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} - \frac{8 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 2 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right)\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{5}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} + \frac{48 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} - \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} - \frac{8 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} + 16 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}{\left(1,2 x + 1 \right)}} - 16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{- \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{48 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 12 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)} - \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(2 x + 1\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{5}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{4}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}^{2} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} + \frac{48 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{32 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{64 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} - \frac{192 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}^{2}} - 96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{96 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}} - \frac{8 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} + \frac{8 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(3,2 x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,2 x + 1 \right)}} - \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,2 x + 1 \right)}} + \frac{128 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} + 16 \cdot 3^{- 2 x} \log{\left(3 \right)} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + \frac{16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\operatorname{polygamma}{\left(1,2 x + 1 \right)}} - 16 \cdot 3^{- 2 x} \Gamma^{2}\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{- \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{3}{\left(0,2 x + 1 \right)}}{\log{\left(3 \right)}} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}^{2}{\left(0,2 x + 1 \right)} - \frac{48 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} \operatorname{polygamma}{\left(1,2 x + 1 \right)}}{\log{\left(3 \right)}} - 12 \cdot 3^{- x} \log{\left(3 \right)} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(0,2 x + 1 \right)} + 24 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(1,2 x + 1 \right)} - \frac{16 \cdot 3^{- x} \Gamma\left(2 x + 1\right) \operatorname{polygamma}{\left(2,2 x + 1 \right)}}{\log{\left(3 \right)}} + 2 \cdot 3^{- x} \log{\left(3 \right)}^{2} \Gamma\left(2 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
$$\lim_{x \to \infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \left(-\infty\right)!$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(2 x\right)!}{3^{x} + 1}\right) = \left(-\infty\right)!$$
Más detalles con x→-oo