Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (1-cos(5*x))/x^2
-
Límite de -6+8*x/3
-
Límite de ((1+x)/(-1+x))^x
-
Límite de (-1+(1+x)*(1+2*x)*(1+3*x))/x
-
Expresiones idénticas
- factorial(x)/sqrt(x)
- factorial(x) dividir por raíz cuadrada de (x)
- factorial(x)/√(x)
- factorialx/sqrtx
- factorial(x) dividir por sqrt(x)
-
Expresiones con funciones
- factorial
- factorial(1+x)/factorial(x)
- factorial(factorial(-1+2*x))/sqrt((2*n)^n)
- factorial(n)^2*Abs(factorial(2*3^(1+n)*(1+n))/(factorial(2*n*3^n)*factorial(1+n)^2))
- factorial(x)^(1/x)
- factorial(x)/factorial(1+x)
- Raíz cuadrada sqrt
- sqrt(x)*(sqrt(2+x)-sqrt(-3+x))
- sqrt(-7+4*x^4+13*x^2)-2*x^2
- sqrt(x)/(100+x)
- sqrt(2+x)/(-4+x)
- sqrt(4+x^2)-sqrt(1+x^2-3*x)
Límite de la función factorial(x)/sqrt(x)
Solución
Ha introducido
[src]
/ x! \
lim |-----|
x->oo| ___|
\\/ x /
$$\lim_{x \to \infty}\left(\frac{x!}{\sqrt{x}}\right)$$
Limit(factorial(x)/sqrt(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
$$\lim_{x \to \infty} \sqrt{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x!}{\sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} \sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 \sqrt{x} \Gamma\left(x + 1\right)}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\sqrt{x}} + \frac{\Gamma\left(x + 1\right)}{2 x^{\frac{3}{2}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{\frac{d}{d x} \left(- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\sqrt{x}} + \frac{\Gamma\left(x + 1\right)}{2 x^{\frac{3}{2}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} + \frac{4 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 24 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 24 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 16 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\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)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{x} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{2}}}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} - 6 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\sqrt{x}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\sqrt{x}} + \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2 x^{\frac{3}{2}}} - \frac{3 \Gamma\left(x + 1\right)}{4 x^{\frac{5}{2}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} + \frac{4 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 24 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 24 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 16 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\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)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{x} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{2}}}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} - 6 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\sqrt{x}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\sqrt{x}} + \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2 x^{\frac{3}{2}}} - \frac{3 \Gamma\left(x + 1\right)}{4 x^{\frac{5}{2}}}}\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
$$\lim_{x \to \infty} \sqrt{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(\frac{x!}{\sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x!}{\frac{d}{d x} \sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 2 \sqrt{x} \Gamma\left(x + 1\right)}{\frac{d}{d x} \frac{1}{\operatorname{polygamma}{\left(0,x + 1 \right)}}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- \frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}}\right)}{\frac{d}{d x} \frac{1}{2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\sqrt{x}} + \frac{\Gamma\left(x + 1\right)}{2 x^{\frac{3}{2}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} + \frac{\Gamma\left(x + 1\right)}{\sqrt{x}}\right)^{2} \left(\frac{\operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 2 \operatorname{polygamma}{\left(0,x + 1 \right)}\right)}{\frac{d}{d x} \left(- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)} - \frac{2 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{\sqrt{x}} + \frac{\Gamma\left(x + 1\right)}{2 x^{\frac{3}{2}}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} + \frac{4 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 24 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 24 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 16 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\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)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{x} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{2}}}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} - 6 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\sqrt{x}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\sqrt{x}} + \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2 x^{\frac{3}{2}}} - \frac{3 \Gamma\left(x + 1\right)}{4 x^{\frac{5}{2}}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{5}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} + \frac{4 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{16 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 24 x \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{4}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - 24 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} + \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{\operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{8 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{\operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{12 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{\operatorname{polygamma}{\left(1,x + 1 \right)}} - 16 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\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)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{4 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{x} + \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(3,x + 1 \right)}}{x \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}^{2}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}^{3}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x \operatorname{polygamma}{\left(1,x + 1 \right)}} - \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{x} - \frac{\Gamma^{2}\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(2,x + 1 \right)}}{x^{2} \operatorname{polygamma}^{2}{\left(1,x + 1 \right)}} + \frac{2 \Gamma^{2}\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{x^{2}}}{- 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}^{3}{\left(0,x + 1 \right)} - 6 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)} \operatorname{polygamma}{\left(1,x + 1 \right)} - 2 \sqrt{x} \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(2,x + 1 \right)} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}^{2}{\left(0,x + 1 \right)}}{\sqrt{x}} - \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(1,x + 1 \right)}}{\sqrt{x}} + \frac{3 \Gamma\left(x + 1\right) \operatorname{polygamma}{\left(0,x + 1 \right)}}{2 x^{\frac{3}{2}}} - \frac{3 \Gamma\left(x + 1\right)}{4 x^{\frac{5}{2}}}}\right)$$
=
$$\infty$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x!}{\sqrt{x}}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x!}{\sqrt{x}}\right) = - \infty i$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x!}{\sqrt{x}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x!}{\sqrt{x}}\right) = 1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x!}{\sqrt{x}}\right) = 1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x!}{\sqrt{x}}\right) = 0$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(\frac{x!}{\sqrt{x}}\right) = - \infty i$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{x!}{\sqrt{x}}\right) = \infty$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(\frac{x!}{\sqrt{x}}\right) = 1$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{x!}{\sqrt{x}}\right) = 1$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{x!}{\sqrt{x}}\right) = 0$$
Más detalles con x→-oo