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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 (4+x^2-5*x)/(8+x^2-6*x)
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Límite de (-3+x^2-2*x)/(-9+x^2)
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Expresiones idénticas
- e^(- uno /x)*log(x^ dos -x)
- e en el grado ( menos 1 dividir por x) multiplicar por logaritmo de (x al cuadrado menos x)
- e en el grado ( menos uno dividir por x) multiplicar por logaritmo de (x en el grado dos menos x)
- e(-1/x)*log(x2-x)
- e-1/x*logx2-x
- e^(-1/x)*log(x²-x)
- e en el grado (-1/x)*log(x en el grado 2-x)
- e^(-1/x)log(x^2-x)
- e(-1/x)log(x2-x)
- e-1/xlogx2-x
- e^-1/xlogx^2-x
- e^(-1 dividir por x)*log(x^2-x)
-
Expresiones semejantes
- e^(-1/x)*log(x^2+x)
- e^(1/x)*log(x^2-x)
-
Expresiones con funciones
- Logaritmo log
- log(cos(2*x))/log(cos(3*x))
- log(x)/(1-x)
- log((5+3*x)/(-4+3*x))^(2*x)
- log(1+2*x)/(3+x)
- log(cos(x))/log(1+x^2)
Límite de la función e^(-1/x)*log(x^2-x)
Solución
Ha introducido
[src]
/ -1 \
| --- |
| x / 2 \|
lim \E *log\x - x//
x->0+
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right)$$
Limit(E^(-1/x)*log(x^2 - x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} e^{- \frac{1}{x}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \left(x - 1\right) \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} e^{- \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\log{\left(x \left(x - 1\right) \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(x - 1\right) e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}^{2}}{x \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}^{2}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{e^{- \frac{1}{x}}}{x}\right)}{\frac{d}{d x} \frac{1}{\log{\left(x \left(x - 1\right) \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \left(x - 1\right) \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right) \log{\left(x \left(x - 1\right) \right)}^{3}}{2 \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right) \log{\left(x \left(x - 1\right) \right)}^{3}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{x \log{\left(x \left(x - 1\right) \right)}^{3}}{2}\right)}{\frac{d}{d x} \frac{1}{\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right)^{2} \left(- \frac{\log{\left(x \left(x - 1\right) \right)}^{3}}{2} - \frac{3 \left(2 x - 1\right) \log{\left(x \left(x - 1\right) \right)}^{2}}{2 \left(x - 1\right)}\right)}{\frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{4 e^{- \frac{1}{x}}}{x^{4}} + \frac{e^{- \frac{1}{x}}}{x^{5}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right)^{2} \left(- \frac{\log{\left(x \left(x - 1\right) \right)}^{3}}{2} - \frac{3 \left(2 x - 1\right) \log{\left(x \left(x - 1\right) \right)}^{2}}{2 \left(x - 1\right)}\right)}{\frac{d}{d x} \left(\frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{4 e^{- \frac{1}{x}}}{x^{4}} + \frac{e^{- \frac{1}{x}}}{x^{5}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{3 \log{\left(x^{2} - x \right)}}{x^{9} e^{\frac{2}{x}} - 2 x^{8} e^{\frac{2}{x}} + x^{7} e^{\frac{2}{x}}} - \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 4 x^{7} e^{\frac{2}{x}} + 2 x^{6} e^{\frac{2}{x}}} + \frac{18 \log{\left(x^{2} - x \right)}}{x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}} + x^{6} e^{\frac{2}{x}}} + \frac{6 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{39 \log{\left(x^{2} - x \right)}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 4 x^{5} e^{\frac{2}{x}} + 2 x^{4} e^{\frac{2}{x}}} + \frac{36 \log{\left(x^{2} - x \right)}}{x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}} + x^{4} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{9} e^{\frac{2}{x}} - x^{8} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}}} - \frac{21 \log{\left(x^{2} - x \right)}^{2}}{x^{8} e^{\frac{2}{x}} - x^{7} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}^{2}}{2 x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}}} + \frac{45 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - x^{6} e^{\frac{2}{x}}} + \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}}} - \frac{36 \log{\left(x^{2} - x \right)}^{2}}{x^{6} e^{\frac{2}{x}} - x^{5} e^{\frac{2}{x}}} - \frac{6 \log{\left(x^{2} - x \right)}^{2}}{2 x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}}} + \frac{9 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - x^{4} e^{\frac{2}{x}}} + \frac{2 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{5}} - \frac{6 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{6}} + \frac{5 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{7}} - \frac{e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{8}}}{- \frac{6 e^{- \frac{1}{x}}}{x^{4}} + \frac{18 e^{- \frac{1}{x}}}{x^{5}} - \frac{9 e^{- \frac{1}{x}}}{x^{6}} + \frac{e^{- \frac{1}{x}}}{x^{7}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{3 \log{\left(x^{2} - x \right)}}{x^{9} e^{\frac{2}{x}} - 2 x^{8} e^{\frac{2}{x}} + x^{7} e^{\frac{2}{x}}} - \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 4 x^{7} e^{\frac{2}{x}} + 2 x^{6} e^{\frac{2}{x}}} + \frac{18 \log{\left(x^{2} - x \right)}}{x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}} + x^{6} e^{\frac{2}{x}}} + \frac{6 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{39 \log{\left(x^{2} - x \right)}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 4 x^{5} e^{\frac{2}{x}} + 2 x^{4} e^{\frac{2}{x}}} + \frac{36 \log{\left(x^{2} - x \right)}}{x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}} + x^{4} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{9} e^{\frac{2}{x}} - x^{8} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}}} - \frac{21 \log{\left(x^{2} - x \right)}^{2}}{x^{8} e^{\frac{2}{x}} - x^{7} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}^{2}}{2 x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}}} + \frac{45 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - x^{6} e^{\frac{2}{x}}} + \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}}} - \frac{36 \log{\left(x^{2} - x \right)}^{2}}{x^{6} e^{\frac{2}{x}} - x^{5} e^{\frac{2}{x}}} - \frac{6 \log{\left(x^{2} - x \right)}^{2}}{2 x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}}} + \frac{9 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - x^{4} e^{\frac{2}{x}}} + \frac{2 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{5}} - \frac{6 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{6}} + \frac{5 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{7}} - \frac{e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{8}}}{- \frac{6 e^{- \frac{1}{x}}}{x^{4}} + \frac{18 e^{- \frac{1}{x}}}{x^{5}} - \frac{9 e^{- \frac{1}{x}}}{x^{6}} + \frac{e^{- \frac{1}{x}}}{x^{7}}}\right)$$
=
$$0$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 4 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} e^{- \frac{1}{x}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \left(x - 1\right) \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} e^{- \frac{1}{x}}}{\frac{d}{d x} \frac{1}{\log{\left(x \left(x - 1\right) \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(x - 1\right) e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}^{2}}{x \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{e^{- \frac{1}{x}} \log{\left(x \left(x - 1\right) \right)}^{2}}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{e^{- \frac{1}{x}}}{x}\right)}{\frac{d}{d x} \frac{1}{\log{\left(x \left(x - 1\right) \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \left(x - 1\right) \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right) \log{\left(x \left(x - 1\right) \right)}^{3}}{2 \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{x \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right) \log{\left(x \left(x - 1\right) \right)}^{3}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{x \log{\left(x \left(x - 1\right) \right)}^{3}}{2}\right)}{\frac{d}{d x} \frac{1}{\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right)^{2} \left(- \frac{\log{\left(x \left(x - 1\right) \right)}^{3}}{2} - \frac{3 \left(2 x - 1\right) \log{\left(x \left(x - 1\right) \right)}^{2}}{2 \left(x - 1\right)}\right)}{\frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{4 e^{- \frac{1}{x}}}{x^{4}} + \frac{e^{- \frac{1}{x}}}{x^{5}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{e^{- \frac{1}{x}}}{x^{2}} - \frac{e^{- \frac{1}{x}}}{x^{3}}\right)^{2} \left(- \frac{\log{\left(x \left(x - 1\right) \right)}^{3}}{2} - \frac{3 \left(2 x - 1\right) \log{\left(x \left(x - 1\right) \right)}^{2}}{2 \left(x - 1\right)}\right)}{\frac{d}{d x} \left(\frac{2 e^{- \frac{1}{x}}}{x^{3}} - \frac{4 e^{- \frac{1}{x}}}{x^{4}} + \frac{e^{- \frac{1}{x}}}{x^{5}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{3 \log{\left(x^{2} - x \right)}}{x^{9} e^{\frac{2}{x}} - 2 x^{8} e^{\frac{2}{x}} + x^{7} e^{\frac{2}{x}}} - \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 4 x^{7} e^{\frac{2}{x}} + 2 x^{6} e^{\frac{2}{x}}} + \frac{18 \log{\left(x^{2} - x \right)}}{x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}} + x^{6} e^{\frac{2}{x}}} + \frac{6 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{39 \log{\left(x^{2} - x \right)}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 4 x^{5} e^{\frac{2}{x}} + 2 x^{4} e^{\frac{2}{x}}} + \frac{36 \log{\left(x^{2} - x \right)}}{x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}} + x^{4} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{9} e^{\frac{2}{x}} - x^{8} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}}} - \frac{21 \log{\left(x^{2} - x \right)}^{2}}{x^{8} e^{\frac{2}{x}} - x^{7} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}^{2}}{2 x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}}} + \frac{45 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - x^{6} e^{\frac{2}{x}}} + \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}}} - \frac{36 \log{\left(x^{2} - x \right)}^{2}}{x^{6} e^{\frac{2}{x}} - x^{5} e^{\frac{2}{x}}} - \frac{6 \log{\left(x^{2} - x \right)}^{2}}{2 x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}}} + \frac{9 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - x^{4} e^{\frac{2}{x}}} + \frac{2 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{5}} - \frac{6 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{6}} + \frac{5 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{7}} - \frac{e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{8}}}{- \frac{6 e^{- \frac{1}{x}}}{x^{4}} + \frac{18 e^{- \frac{1}{x}}}{x^{5}} - \frac{9 e^{- \frac{1}{x}}}{x^{6}} + \frac{e^{- \frac{1}{x}}}{x^{7}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{3 \log{\left(x^{2} - x \right)}}{x^{9} e^{\frac{2}{x}} - 2 x^{8} e^{\frac{2}{x}} + x^{7} e^{\frac{2}{x}}} - \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 4 x^{7} e^{\frac{2}{x}} + 2 x^{6} e^{\frac{2}{x}}} + \frac{18 \log{\left(x^{2} - x \right)}}{x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}} + x^{6} e^{\frac{2}{x}}} + \frac{6 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{39 \log{\left(x^{2} - x \right)}}{x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}} + x^{5} e^{\frac{2}{x}}} - \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 4 x^{5} e^{\frac{2}{x}} + 2 x^{4} e^{\frac{2}{x}}} + \frac{36 \log{\left(x^{2} - x \right)}}{x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}} + x^{4} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}}{x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}} + x^{3} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{x^{9} e^{\frac{2}{x}} - x^{8} e^{\frac{2}{x}}} + \frac{3 \log{\left(x^{2} - x \right)}^{2}}{2 x^{8} e^{\frac{2}{x}} - 2 x^{7} e^{\frac{2}{x}}} - \frac{21 \log{\left(x^{2} - x \right)}^{2}}{x^{8} e^{\frac{2}{x}} - x^{7} e^{\frac{2}{x}}} - \frac{12 \log{\left(x^{2} - x \right)}^{2}}{2 x^{7} e^{\frac{2}{x}} - 2 x^{6} e^{\frac{2}{x}}} + \frac{45 \log{\left(x^{2} - x \right)}^{2}}{x^{7} e^{\frac{2}{x}} - x^{6} e^{\frac{2}{x}}} + \frac{15 \log{\left(x^{2} - x \right)}^{2}}{2 x^{6} e^{\frac{2}{x}} - 2 x^{5} e^{\frac{2}{x}}} - \frac{36 \log{\left(x^{2} - x \right)}^{2}}{x^{6} e^{\frac{2}{x}} - x^{5} e^{\frac{2}{x}}} - \frac{6 \log{\left(x^{2} - x \right)}^{2}}{2 x^{5} e^{\frac{2}{x}} - 2 x^{4} e^{\frac{2}{x}}} + \frac{9 \log{\left(x^{2} - x \right)}^{2}}{x^{5} e^{\frac{2}{x}} - x^{4} e^{\frac{2}{x}}} + \frac{2 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{5}} - \frac{6 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{6}} + \frac{5 e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{7}} - \frac{e^{- \frac{2}{x}} \log{\left(x^{2} - x \right)}^{3}}{x^{8}}}{- \frac{6 e^{- \frac{1}{x}}}{x^{4}} + \frac{18 e^{- \frac{1}{x}}}{x^{5}} - \frac{9 e^{- \frac{1}{x}}}{x^{6}} + \frac{e^{- \frac{1}{x}}}{x^{7}}}\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
$$\lim_{x \to 0^-}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = 0$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = -\infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = \infty$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = 0$$
$$\lim_{x \to \infty}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = \infty$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = -\infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right) = \infty$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ -1 \
| --- |
| x / 2 \|
lim \E *log\x - x//
x->0+
$$\lim_{x \to 0^+}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right)$$
0
$$0$$
= (2.05635159657357e-27 + 4.19828910685802e-29j)
/ -1 \
| --- |
| x / 2 \|
lim \E *log\x - x//
x->0-
$$\lim_{x \to 0^-}\left(e^{- \frac{1}{x}} \log{\left(x^{2} - x \right)}\right)$$
-oo
$$-\infty$$
= 0.118105343906953
= 0.118105343906953
Respuesta numérica
[src]
(2.05635159657357e-27 + 4.19828910685802e-29j)
(2.05635159657357e-27 + 4.19828910685802e-29j)