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Expresiones idénticas
- log(cinco *x)/log(sin(x))+log(cos(x))
- logaritmo de (5 multiplicar por x) dividir por logaritmo de ( seno de (x)) más logaritmo de ( coseno de (x))
- logaritmo de (cinco multiplicar por x) dividir por logaritmo de ( seno de (x)) más logaritmo de ( coseno de (x))
- log(5x)/log(sin(x))+log(cos(x))
- log5x/logsinx+logcosx
- log(5*x) dividir por log(sin(x))+log(cos(x))
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Expresiones semejantes
- log(5*x)/log(sin(x))-log(cos(x))
- log(5*x)/log(sinx)+log(cosx)
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Expresiones con funciones
- Logaritmo log
- log(x*(1+sqrt(1+2*x))^2/2)
- log(sqrt(3+x^2)/x)
- log(x)^3-3*log(x)/x
- log(-8+x^2)/(-3+x)
- log(1+x/n)
- Logaritmo log
- log(x*(1+sqrt(1+2*x))^2/2)
- log(sqrt(3+x^2)/x)
- log(x)^3-3*log(x)/x
- log(-8+x^2)/(-3+x)
- log(1+x/n)
- Seno sin
- sin(x)^5/x^3
- sin(x/(x-i))
- sin(2*x)^3*tan(3*x)/x^2
- sin(10+x)^2/(-8+sqrt(-36+x^2))
- sin(4*a)/x
- Logaritmo log
- log(x*(1+sqrt(1+2*x))^2/2)
- log(sqrt(3+x^2)/x)
- log(x)^3-3*log(x)/x
- log(-8+x^2)/(-3+x)
- log(1+x/n)
- Coseno cos
- cos(2*x+pi/3)/sin(3*x)
- cos(x)^3+sin(x)^2
- cos(x)^2/sin(2*x)
- cos(-cos(x)+2*x)/(1-x*cos(4))
- cos(pi/(3*n))
Límite de la función log(5*x)/log(sin(x))+log(cos(x))
Solución
Ha introducido
[src]
/ log(5*x) \ lim |----------- + log(cos(x))| x->0+\log(sin(x)) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right)$$
Limit(log(5*x)/log(sin(x)) + log(cos(x)), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}\right)^{2} \cos{\left(x \right)}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(x \right)}^{2} + 2 \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + 2 \log{\left(5 \right)} \log{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}^{2} \log{\left(\cos{\left(x \right)} \right)}^{2} + 2 \log{\left(5 \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + \log{\left(5 \right)}^{2}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(x \right)}^{2} + 2 \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + 2 \log{\left(5 \right)} \log{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}^{2} \log{\left(\cos{\left(x \right)} \right)}^{2} + 2 \log{\left(5 \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + \log{\left(5 \right)}^{2}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right)$$
=
Introducimos una pequeña modificación de la función bajo el signo del límite
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(5 x \right)} + \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)}\right)^{2} \cos{\left(x \right)}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(x \right)}^{2} + 2 \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + 2 \log{\left(5 \right)} \log{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}^{2} \log{\left(\cos{\left(x \right)} \right)}^{2} + 2 \log{\left(5 \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + \log{\left(5 \right)}^{2}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\log{\left(x \right)}^{2} + 2 \log{\left(x \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + 2 \log{\left(5 \right)} \log{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)}^{2} \log{\left(\cos{\left(x \right)} \right)}^{2} + 2 \log{\left(5 \right)} \log{\left(\sin{\left(x \right)} \right)} \log{\left(\cos{\left(x \right)} \right)} + \log{\left(5 \right)}^{2}}{\left(\frac{\log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin{\left(x \right)}} - \frac{1}{x}\right) \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 1 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \log{\left(\left\langle -1, 1\right\rangle \right)} + \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \frac{\log{\left(\sin{\left(1 \right)} \right)} \log{\left(\cos{\left(1 \right)} \right)} + \log{\left(5 \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \frac{\log{\left(\sin{\left(1 \right)} \right)} \log{\left(\cos{\left(1 \right)} \right)} + \log{\left(5 \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \log{\left(\left\langle -1, 1\right\rangle \right)} + \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \log{\left(\left\langle -1, 1\right\rangle \right)} + \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \frac{\log{\left(\sin{\left(1 \right)} \right)} \log{\left(\cos{\left(1 \right)} \right)} + \log{\left(5 \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \frac{\log{\left(\sin{\left(1 \right)} \right)} \log{\left(\cos{\left(1 \right)} \right)} + \log{\left(5 \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right) = \log{\left(\left\langle -1, 1\right\rangle \right)} + \frac{\infty}{\log{\left(\left\langle -1, 1\right\rangle \right)}}$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/ log(5*x) \ lim |----------- + log(cos(x))| x->0+\log(sin(x)) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right)$$
1
$$1$$
= 0.813728509301446
/ log(5*x) \ lim |----------- + log(cos(x))| x->0-\log(sin(x)) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(5 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}} + \log{\left(\cos{\left(x \right)} \right)}\right)$$
1
$$1$$
= (0.836922820934514 - 0.0598547353136279j)
= (0.836922820934514 - 0.0598547353136279j)