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- Límite de la función:
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Límite de (-5+x)/(-25+x^2)
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Límite de (-1+e^x)/x
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Límite de (x^2-2*x)/(4+x^2-4*x)
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Límite de (1+x)^(1/x)
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Expresiones idénticas
- log(sin(tres *x))/log(x)
- logaritmo de ( seno de (3 multiplicar por x)) dividir por logaritmo de (x)
- logaritmo de ( seno de (tres multiplicar por x)) dividir por logaritmo de (x)
- log(sin(3x))/log(x)
- logsin3x/logx
- log(sin(3*x)) dividir por log(x)
-
Expresiones semejantes
- log(sin(3*x))/(log(x)*sin(x))
-
Expresiones con funciones
- Logaritmo log
- log(x)*sin(x)
- log(1/x)^x
- log(x)/(1+2*log(sin(x)))
- log(x)/cot(2*x)
- log(x)*log(1-x)
- Seno sin
- sin(2*x)/x
- sin(3*x)/sin(2*x)
- sin(3*x)/(4*x)
- sin(x)^2/x
- sin(x)/(2*x)
- Logaritmo log
- log(x)*sin(x)
- log(1/x)^x
- log(x)/(1+2*log(sin(x)))
- log(x)/cot(2*x)
- log(x)*log(1-x)
Límite de la función log(sin(3*x))/log(x)
Solución
Ha introducido
[src]
/log(sin(3*x))\ lim |-------------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right)$$
Limit(log(sin(3*x))/log(x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{5}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{18 x^{4} \log{\left(x \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{6 x^{4} \log{\left(x \right)}^{4}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{6}}}{\frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{5}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{18 x^{4} \log{\left(x \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{6 x^{4} \log{\left(x \right)}^{4}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{6}}}{\frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
0/0,
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 x \log{\left(x \right)}^{2}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(x \right)}^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{5}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{18 x^{4} \log{\left(x \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{6 x^{4} \log{\left(x \right)}^{4}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{6}}}{\frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{5}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{18 x^{4} \log{\left(x \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{6 x^{4} \log{\left(x \right)}^{4}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{5}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{3 x^{4} \log{\left(x \right)}^{6}}}{\frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = \infty$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = -\infty$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right) = 0$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/log(sin(3*x))\ lim |-------------| x->0+\ log(x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right)$$
1
$$1$$
= 0.872760621735037
/log(sin(3*x))\ lim |-------------| x->0-\ log(x) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(x \right)}}\right)$$
1
$$1$$
= (0.888785968782996 - 0.0416375248293838j)
= (0.888785968782996 - 0.0416375248293838j)
Gráfico