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Límite de (-1+e^x)/x
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Límite de x^2*log(x)
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Límite de (3-sqrt(5+x))/(1-sqrt(5-x))
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Límite de x^2
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Expresiones idénticas
- ((uno +x)^(uno /x)-e)^ dos /(-x+log(x+cos(x)))
- ((1 más x) en el grado (1 dividir por x) menos e) al cuadrado dividir por ( menos x más logaritmo de (x más coseno de (x)))
- ((uno más x) en el grado (uno dividir por x) menos e) en el grado dos dividir por ( menos x más logaritmo de (x más coseno de (x)))
- ((1+x)(1/x)-e)2/(-x+log(x+cos(x)))
- 1+x1/x-e2/-x+logx+cosx
- ((1+x)^(1/x)-e)²/(-x+log(x+cos(x)))
- ((1+x) en el grado (1/x)-e) en el grado 2/(-x+log(x+cos(x)))
- 1+x^1/x-e^2/-x+logx+cosx
- ((1+x)^(1 dividir por x)-e)^2 dividir por (-x+log(x+cos(x)))
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Expresiones semejantes
- ((1-x)^(1/x)-e)^2/(-x+log(x+cos(x)))
- ((1+x)^(1/x)-e)^2/(x+log(x+cos(x)))
- ((1+x)^(1/x)-e)^2/(-x+log(x-cos(x)))
- ((1+x)^(1/x)+e)^2/(-x+log(x+cos(x)))
- ((1+x)^(1/x)-e)^2/(-x-log(x+cos(x)))
- ((1+x)^(1/x)-e)^2/(-x+log(x+cosx))
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Expresiones con funciones
- Logaritmo log
- log(x)/(1-x^2)
- log(sin(x))
- log(1-x)/x
- log(-2+x)
- log(2+x)
- Coseno cos
- cos(x)/x^2
- cos(3*x)*sin(2*x)/(cot(4*x)*sin(x))
- cos(pi*x)^(1/(x*sin(pi*x)))
- cos(5*x)
- cos(5*x)*sin(2*x)/tan(x)
Límite de la función ((1+x)^(1/x)-e)^2/(-x+log(x+cos(x)))
Solución
Ha introducido
[src]
/ 2 \
| /x _______ \ |
| \\/ 1 + x - E/ |
lim |--------------------|
x->0+\-x + log(x + cos(x))/
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
Limit(((1 + x)^(1/x) - E)^2/(-x + log(x + cos(x))), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(\left(x + 1\right)^{\frac{2}{x}} - 2 e \left(x + 1\right)^{\frac{1}{x}} + e^{2}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(- x + \log{\left(x + \cos{\left(x \right)} \right)}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\left(x + 1\right)^{\frac{2}{x}} - 2 e \left(x + 1\right)^{\frac{1}{x}} + e^{2}\right)}{\frac{d}{d x} \left(- x + \log{\left(x + \cos{\left(x \right)} \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{2} + x} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{2} + x} - \frac{2 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{2}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{2}}}{-1 - \frac{\sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{1}{x + \cos{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{2} + x} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{2} + x} - \frac{2 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{2}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{2}}\right)}{\frac{d}{d x} \left(-1 - \frac{\sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{1}{x + \cos{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 x \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{4 e x \left(x + 1\right)^{\frac{1}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} - \frac{8 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} - \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{3} + x^{2}} + \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{3} + x^{2}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{3}} - \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{3}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}}}{- \frac{\sin^{2}{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{1}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x + \cos{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 x \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{4 e x \left(x + 1\right)^{\frac{1}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} - \frac{8 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} - \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{3} + x^{2}} + \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{3} + x^{2}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{3}} - \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{3}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}}}{- \frac{\sin^{2}{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{1}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x + \cos{\left(x \right)}}}\right)$$
=
$$- \frac{e^{2}}{4}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+}\left(\left(x + 1\right)^{\frac{2}{x}} - 2 e \left(x + 1\right)^{\frac{1}{x}} + e^{2}\right) = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+}\left(- x + \log{\left(x + \cos{\left(x \right)} \right)}\right) = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\left(x + 1\right)^{\frac{2}{x}} - 2 e \left(x + 1\right)^{\frac{1}{x}} + e^{2}\right)}{\frac{d}{d x} \left(- x + \log{\left(x + \cos{\left(x \right)} \right)}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{2} + x} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{2} + x} - \frac{2 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{2}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{2}}}{-1 - \frac{\sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{1}{x + \cos{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{2} + x} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{2} + x} - \frac{2 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{2}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{2}}\right)}{\frac{d}{d x} \left(-1 - \frac{\sin{\left(x \right)}}{x + \cos{\left(x \right)}} + \frac{1}{x + \cos{\left(x \right)}}\right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 x \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{4 e x \left(x + 1\right)^{\frac{1}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} - \frac{8 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} - \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{3} + x^{2}} + \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{3} + x^{2}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{3}} - \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{3}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}}}{- \frac{\sin^{2}{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{1}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x + \cos{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{4 x \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{4 e x \left(x + 1\right)^{\frac{1}{x}}}{x^{4} + 2 x^{3} + x^{2}} + \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{4} + 2 x^{3} + x^{2}} - \frac{8 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} - \frac{2 \left(x + 1\right)^{\frac{2}{x}}}{x^{3} + x^{2}} + \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{4} + x^{3}} + \frac{2 e \left(x + 1\right)^{\frac{1}{x}}}{x^{3} + x^{2}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}}{x^{3}} - \frac{4 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}}{x^{3}} + \frac{4 \left(x + 1\right)^{\frac{2}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}} - \frac{2 e \left(x + 1\right)^{\frac{1}{x}} \log{\left(x + 1 \right)}^{2}}{x^{4}}}{- \frac{\sin^{2}{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} + \frac{2 \sin{\left(x \right)}}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{1}{x^{2} + 2 x \cos{\left(x \right)} + \cos^{2}{\left(x \right)}} - \frac{\cos{\left(x \right)}}{x + \cos{\left(x \right)}}}\right)$$
=
$$- \frac{e^{2}}{4}$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 2 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/ 2 \
| /x _______ \ |
| \\/ 1 + x - E/ |
lim |--------------------|
x->0+\-x + log(x + cos(x))/
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
2 -e ---- 4
$$- \frac{e^{2}}{4}$$
= -1.84726402473266
/ 2 \
| /x _______ \ |
| \\/ 1 + x - E/ |
lim |--------------------|
x->0-\-x + log(x + cos(x))/
$$\lim_{x \to 0^-}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
2 -e ---- 4
$$- \frac{e^{2}}{4}$$
= -1.84726402473266
= -1.84726402473266
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = - \frac{e^{2}}{4}$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = - \frac{e^{2}}{4}$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = \frac{- 4 e + 4 + e^{2}}{-1 + \log{\left(\cos{\left(1 \right)} + 1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = \frac{- 4 e + 4 + e^{2}}{-1 + \log{\left(\cos{\left(1 \right)} + 1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = - \frac{e^{2}}{4}$$
$$\lim_{x \to \infty}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right)$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = \frac{- 4 e + 4 + e^{2}}{-1 + \log{\left(\cos{\left(1 \right)} + 1 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = \frac{- 4 e + 4 + e^{2}}{-1 + \log{\left(\cos{\left(1 \right)} + 1 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\left(\left(x + 1\right)^{\frac{1}{x}} - e\right)^{2}}{- x + \log{\left(x + \cos{\left(x \right)} \right)}}\right) = 0$$
Más detalles con x→-oo