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Límite de (-9+x^2)/(3+x)
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Límite de x^2/(1-cos(6*x))
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Límite de (-3+sqrt(1+2*x))/(sqrt(-2+x)-sqrt(2))
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Límite de (4+x^2-5*x)/(8+x^2-6*x)
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Expresiones idénticas
- dos ^(-x)*x^(sqrt(x))
- 2 en el grado ( menos x) multiplicar por x en el grado ( raíz cuadrada de (x))
- dos en el grado ( menos x) multiplicar por x en el grado ( raíz cuadrada de (x))
- 2^(-x)*x^(√(x))
- 2(-x)*x(sqrt(x))
- 2-x*xsqrtx
- 2^(-x)x^(sqrt(x))
- 2(-x)x(sqrt(x))
- 2-xxsqrtx
- 2^-xx^sqrtx
-
Expresiones semejantes
- 2^(x)*x^(sqrt(x))
-
Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(2+x^2+4*x)-sqrt(2+x^2-2*x)
- sqrt(-1+x^2)-x
- sqrt(n+n^2)-n
- sqrt(2+n)/sqrt(1+n)
- sqrt(2+x)-(20+x)^(1/3)
Límite de la función 2^(-x)*x^(sqrt(x))
Solución
Ha introducido
[src]
/ ___\
| -x \/ x |
lim \2 *x /
x->oo
$$\lim_{x \to \infty}\left(2^{- x} x^{\sqrt{x}}\right)$$
Limit(2^(-x)*x^(sqrt(x)), x, oo, dir='-')
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to \infty} x^{\sqrt{x}} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 2^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(2^{- x} x^{\sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{\sqrt{x}}}{\frac{d}{d x} 2^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{- x} x^{\sqrt{x}} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)}{\log{\left(2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)}{\frac{d}{d x} 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\log{\left(x \right)}}{4 x^{\frac{3}{2}} \left(2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{\sqrt{x}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{\sqrt{x}}}}{\frac{d}{d x} \left(- \frac{4 x^{\frac{3}{2}}}{\log{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}}\right)}{\frac{d}{d x} \left(- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} - 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{4} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} - \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{8 x^{\frac{5}{2}}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{4 x^{\frac{5}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{12 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} + 12 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} - \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{2}} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{3}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} - \frac{10 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{2}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{3}} + \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} - 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{4} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} - \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{8 x^{\frac{5}{2}}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{4 x^{\frac{5}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{12 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} + 12 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} - \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{2}} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{3}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} - \frac{10 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{2}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{3}} + \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}^{3}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty} x^{\sqrt{x}} = \infty$$
y el límite para el denominador es
$$\lim_{x \to \infty} 2^{x} = \infty$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to \infty}\left(2^{- x} x^{\sqrt{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x^{\sqrt{x}}}{\frac{d}{d x} 2^{x}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{2^{- x} x^{\sqrt{x}} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)}{\log{\left(2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}\right)}{\frac{d}{d x} 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}\right)$$
=
$$\lim_{x \to \infty}\left(- \frac{\log{\left(x \right)}}{4 x^{\frac{3}{2}} \left(2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{\sqrt{x}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \frac{1}{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{\sqrt{x}}}}{\frac{d}{d x} \left(- \frac{4 x^{\frac{3}{2}}}{\log{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(- 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}}\right)}{\frac{d}{d x} \left(- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}}\right)}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} - 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{4} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} - \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{8 x^{\frac{5}{2}}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{4 x^{\frac{5}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{12 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} + 12 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} - \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{2}} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{3}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} - \frac{10 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{2}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{3}} + \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}^{3}}}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{- 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} - 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{4} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2}}{4 x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{x} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2}}{x} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}^{2}}{4 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{2 x^{2}} + \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} + \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} + \frac{2 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{2^{x} x^{- \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{2 \sqrt{x}} - \frac{1}{\sqrt{x}}\right) \log{\left(2 \right)} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} - \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{3 \cdot 2^{x} x^{- \sqrt{x}} \log{\left(2 \right)} \log{\left(x \right)}}{8 x^{\frac{5}{2}}} - \frac{2^{x} x^{- \sqrt{x}} \log{\left(2 \right)}}{4 x^{\frac{5}{2}}}}{- \frac{6 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{12 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} \sqrt{x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{5}}{\log{\left(x \right)}^{2}} + 6 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{3}}{\log{\left(x \right)}^{2}} + 12 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}} - \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{2}} + \frac{16 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{x \log{\left(x \right)}^{3}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2} \log{\left(x \right)}}{2 \sqrt{x}} - \frac{5 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x}} - \frac{2 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}} + \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \left(- \frac{\log{\left(x \right)}}{\sqrt{x}} - \frac{2}{\sqrt{x}}\right) \log{\left(2 \right)}^{2}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3} \log{\left(x \right)}}{\sqrt{x}} - \frac{10 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x}} - \frac{4 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}} - \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{2}} + \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{3}}{\sqrt{x} \log{\left(x \right)}^{2}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{4}}{\sqrt{x} \log{\left(x \right)}^{3}} + \frac{3 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2} \log{\left(x \right)}}{4 x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}}} + \frac{2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}} - \frac{8 \cdot 2^{2 x} x^{- 2 \sqrt{x}} \log{\left(2 \right)}^{2}}{x^{\frac{3}{2}} \log{\left(x \right)}^{3}}}\right)$$
=
$$0$$
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
$$\lim_{x \to \infty}\left(2^{- x} x^{\sqrt{x}}\right) = 0$$
$$\lim_{x \to 0^-}\left(2^{- x} x^{\sqrt{x}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(2^{- x} x^{\sqrt{x}}\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(2^{- x} x^{\sqrt{x}}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(2^{- x} x^{\sqrt{x}}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(2^{- x} x^{\sqrt{x}}\right) = \infty$$
Más detalles con x→-oo
$$\lim_{x \to 0^-}\left(2^{- x} x^{\sqrt{x}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(2^{- x} x^{\sqrt{x}}\right) = 1$$
Más detalles con x→0 a la derecha
$$\lim_{x \to 1^-}\left(2^{- x} x^{\sqrt{x}}\right) = \frac{1}{2}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(2^{- x} x^{\sqrt{x}}\right) = \frac{1}{2}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(2^{- x} x^{\sqrt{x}}\right) = \infty$$
Más detalles con x→-oo