Sr Examen
Otras calculadoras:
-
¿Cómo usar?
- Límite de la función:
-
Límite de (-9+x^2)/(3+x)
-
Límite de x^2/(1-cos(6*x))
-
Límite de (4+x^2-5*x)/(8+x^2-6*x)
-
Límite de (-3+x^2-2*x)/(-9+x^2)
-
Expresiones idénticas
- log(sin(tres *x))/log(sin(x))
- logaritmo de ( seno de (3 multiplicar por x)) dividir por logaritmo de ( seno de (x))
- logaritmo de ( seno de (tres multiplicar por x)) dividir por logaritmo de ( seno de (x))
- log(sin(3x))/log(sin(x))
- logsin3x/logsinx
- log(sin(3*x)) dividir por log(sin(x))
-
Expresiones semejantes
- log(sin(3*x))/log(sinx)
-
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)
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x^2)/x
- sin(x)/|x|
- sin(x)/asin(x)
- 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)
- Seno sin
- sin(5*x)/sin(2*x)
- sin(3*x)/(3*x)
- sin(x^2)/x
- sin(x)/|x|
- sin(x)/asin(x)
Límite de la función log(sin(3*x))/log(sin(x))
Solución
Ha introducido
[src]
/log(sin(3*x))\ lim |-------------| x->0+\ log(sin(x)) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
Limit(log(sin(3*x))/log(sin(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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{18 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{6 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{8 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{18 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{6 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{8 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}} = 0$$
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.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(x \right)} \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}\right) \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} - \frac{\sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{18 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{6 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{8 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{18 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{2 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{2}{\left(x \right)}} + \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{6 \log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} + \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos{\left(x \right)} \cos^{2}{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{3}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{5} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{3} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{9 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos^{2}{\left(x \right)} \cos{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{6} \sin^{4}{\left(x \right)}}}{\frac{8 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{2} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)}}{3 \log{\left(\sin{\left(x \right)} \right)}^{3} \sin{\left(x \right)}} + \frac{4 \cos{\left(x \right)} \cos{\left(3 x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{2}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{3} \sin^{3}{\left(x \right)}} - \frac{2 \sin{\left(3 x \right)} \cos^{2}{\left(x \right)}}{\log{\left(\sin{\left(x \right)} \right)}^{4} \sin^{3}{\left(x \right)}}}\right)$$
=
$$1$$
Como puedes ver, hemos aplicado el método de l'Hopital (utilizando la derivada del numerador y denominador) 3 vez (veces)
Gráfica
A la izquierda y a la derecha
[src]
/log(sin(3*x))\ lim |-------------| x->0+\ log(sin(x)) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
1
$$1$$
= 0.87293274500189
/log(sin(3*x))\ lim |-------------| x->0-\ log(sin(x)) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right)$$
1
$$1$$
= (0.888939992020216 - 0.0415653426978691j)
= (0.888939992020216 - 0.0415653426978691j)
Otros límites con x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)}}{\log{\left(\sin{\left(1 \right)} \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{\log{\left(\sin{\left(x \right)} \right)}}\right) = 1$$
Más detalles con x→-oo
Gráfico