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Límite de la función (1+log(sin(3*x)))/log(2*x)
Solución
Ha introducido
[src]
/1 + log(sin(3*x))\ lim |-----------------| x->0+\ log(2*x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right)$$
Limit((1 + log(sin(3*x)))/log(2*x), x, 0)
Método de l'Hopital
Tenemos la indeterminación de tipo
tal que el límite para el numerador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(2 x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)} + 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(2 x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)} + 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2} \sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2} \sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2}}}{\frac{d}{d x} \frac{1}{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \left(\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}\right) \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \left(\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}\right) \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \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} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{4 \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} + 18 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 60 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 18 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{x^{4} \log{\left(x \right)}^{6} + 6 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 15 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 20 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 15 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + x^{4} \log{\left(2 \right)}^{6}} - \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} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} + \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} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{8 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \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} + 72 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 108 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 72 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 18 x^{4} \log{\left(2 \right)}^{4}} - \frac{2 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{4} + 36 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 54 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 36 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{4}} - \frac{\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)}^{4} + 12 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{2 x^{4} \log{\left(x \right)}^{4} + 8 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{4} \log{\left(2 \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} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \frac{4 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \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)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{9 \log{\left(\sin{\left(3 x \right)} \right)} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}} - \frac{2 \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}}}{- \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{3}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3} + 3 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2} + 6 x^{3} \log{\left(2 \right)} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2} + 2 x^{2} \log{\left(2 \right)} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{2}} + \frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2} + 2 x \log{\left(2 \right)} \log{\left(x \right)} + x \log{\left(2 \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \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} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{4 \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} + 18 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 60 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 18 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{x^{4} \log{\left(x \right)}^{6} + 6 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 15 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 20 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 15 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + x^{4} \log{\left(2 \right)}^{6}} - \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} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} + \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} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{8 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \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} + 72 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 108 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 72 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 18 x^{4} \log{\left(2 \right)}^{4}} - \frac{2 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{4} + 36 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 54 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 36 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{4}} - \frac{\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)}^{4} + 12 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{2 x^{4} \log{\left(x \right)}^{4} + 8 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{4} \log{\left(2 \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} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \frac{4 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \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)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{9 \log{\left(\sin{\left(3 x \right)} \right)} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}} - \frac{2 \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}}}{- \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{3}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3} + 3 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2} + 6 x^{3} \log{\left(2 \right)} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2} + 2 x^{2} \log{\left(2 \right)} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{2}} + \frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2} + 2 x \log{\left(2 \right)} \log{\left(x \right)} + x \log{\left(2 \right)}^{2}}}\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
$$\lim_{x \to 0^+} \frac{1}{\log{\left(2 x \right)}} = 0$$
y el límite para el denominador es
$$\lim_{x \to 0^+} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)} + 1} = 0$$
Vamos a probar las derivadas del numerador y denominador hasta eliminar la indeterminación.
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{1}{\log{\left(2 x \right)}}}{\frac{d}{d x} \frac{1}{\log{\left(\sin{\left(3 x \right)} \right)} + 1}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2} \sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2} \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2} \sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \frac{\sin{\left(3 x \right)}}{3 x \log{\left(2 x \right)}^{2}}}{\frac{d}{d x} \frac{1}{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \left(\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}\right) \sin{\left(3 x \right)}}{6 \cos{\left(3 x \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \left(\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}\right) \sin{\left(3 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(- \frac{\left(\log{\left(\sin{\left(3 x \right)} \right)} + 1\right)^{3} \sin{\left(3 x \right)}}{6}\right)}{\frac{d}{d x} \frac{1}{\frac{\cos{\left(3 x \right)}}{x \log{\left(2 x \right)}^{2}} - \frac{\sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{2}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{2} \log{\left(2 x \right)}^{3}}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \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} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{4 \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} + 18 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 60 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 18 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{x^{4} \log{\left(x \right)}^{6} + 6 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 15 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 20 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 15 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + x^{4} \log{\left(2 \right)}^{6}} - \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} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} + \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} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{8 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \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} + 72 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 108 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 72 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 18 x^{4} \log{\left(2 \right)}^{4}} - \frac{2 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{4} + 36 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 54 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 36 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{4}} - \frac{\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)}^{4} + 12 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{2 x^{4} \log{\left(x \right)}^{4} + 8 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{4} \log{\left(2 \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} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \frac{4 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \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)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{9 \log{\left(\sin{\left(3 x \right)} \right)} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}} - \frac{2 \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}}}{- \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{3}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3} + 3 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2} + 6 x^{3} \log{\left(2 \right)} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2} + 2 x^{2} \log{\left(2 \right)} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{2}} + \frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2} + 2 x \log{\left(2 \right)} \log{\left(x \right)} + x \log{\left(2 \right)}^{2}}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{- \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} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{6} + 54 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 135 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 180 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 135 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 54 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{6}} - \frac{4 \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} + 18 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 60 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 18 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{6}} - \frac{2 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{x^{4} \log{\left(x \right)}^{6} + 6 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{5} + 15 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{4} + 20 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{3} + 15 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)}^{2} + 6 x^{4} \log{\left(2 \right)}^{5} \log{\left(x \right)} + x^{4} \log{\left(2 \right)}^{6}} - \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} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} - \frac{8 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{5} + 45 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 90 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 90 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 45 x^{4} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{5}} + \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} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{12 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{18 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{5}} + \frac{8 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{5} + 15 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{4} + 30 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{3} + 30 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)}^{2} + 15 x^{3} \log{\left(2 \right)}^{4} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \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} + 72 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 108 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 72 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 18 x^{4} \log{\left(2 \right)}^{4}} - \frac{2 \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{9 x^{4} \log{\left(x \right)}^{4} + 36 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 54 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 36 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 9 x^{4} \log{\left(2 \right)}^{4}} - \frac{\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)}^{4} + 12 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{4} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)} \sin^{2}{\left(3 x \right)} \cos{\left(3 x \right)}}{2 x^{4} \log{\left(x \right)}^{4} + 8 x^{4} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{4} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{4} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{4} \log{\left(2 \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} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \frac{4 \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{4} + 12 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 18 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 12 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{4}} + \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)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} + \frac{3 \log{\left(\sin{\left(3 x \right)} \right)} \sin{\left(3 x \right)} \cos^{2}{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{\log{\left(\sin{\left(3 x \right)} \right)}^{3} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{9 \log{\left(\sin{\left(3 x \right)} \right)} \cos^{3}{\left(3 x \right)}}{2 x^{2} \log{\left(x \right)}^{4} + 8 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 12 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 8 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + 2 x^{2} \log{\left(2 \right)}^{4}} - \frac{3 \log{\left(\sin{\left(3 x \right)} \right)}^{2} \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}} - \frac{2 \cos^{3}{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{4} + 4 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{2} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{4}}}{- \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{4} + 4 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{3} + 6 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)}^{2} + 4 x^{3} \log{\left(2 \right)}^{3} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{4}} - \frac{2 \sin{\left(3 x \right)}}{x^{3} \log{\left(x \right)}^{3} + 3 x^{3} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{3} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{3} \log{\left(2 \right)}^{3}} + \frac{4 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{3} + 3 x^{2} \log{\left(2 \right)} \log{\left(x \right)}^{2} + 3 x^{2} \log{\left(2 \right)}^{2} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{3}} - \frac{2 \sin{\left(3 x \right)}}{3 x^{3} \log{\left(x \right)}^{2} + 6 x^{3} \log{\left(2 \right)} \log{\left(x \right)} + 3 x^{3} \log{\left(2 \right)}^{2}} + \frac{2 \cos{\left(3 x \right)}}{x^{2} \log{\left(x \right)}^{2} + 2 x^{2} \log{\left(2 \right)} \log{\left(x \right)} + x^{2} \log{\left(2 \right)}^{2}} + \frac{3 \sin{\left(3 x \right)}}{x \log{\left(x \right)}^{2} + 2 x \log{\left(2 \right)} \log{\left(x \right)} + x \log{\left(2 \right)}^{2}}}\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
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 1$$
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)} + 1}{\log{\left(2 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)} + 1}{\log{\left(2 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 0$$
Más detalles con x→-oo
Más detalles con x→0 a la izquierda
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 0$$
Más detalles con x→oo
$$\lim_{x \to 1^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)} + 1}{\log{\left(2 \right)}}$$
Más detalles con x→1 a la izquierda
$$\lim_{x \to 1^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = \frac{\log{\left(\sin{\left(3 \right)} \right)} + 1}{\log{\left(2 \right)}}$$
Más detalles con x→1 a la derecha
$$\lim_{x \to -\infty}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right) = 0$$
Más detalles con x→-oo
A la izquierda y a la derecha
[src]
/1 + log(sin(3*x))\ lim |-----------------| x->0+\ log(2*x) /
$$\lim_{x \to 0^+}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right)$$
1
$$1$$
/1 + log(sin(3*x))\ lim |-----------------| x->0-\ log(2*x) /
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\sin{\left(3 x \right)} \right)} + 1}{\log{\left(2 x \right)}}\right)$$
1
$$1$$
= (0.848851942155515 - 0.0621303850488519j)
= (0.848851942155515 - 0.0621303850488519j)