$$\lim_{x \to \infty} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \log{\left(\frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(3 \right)}} \right)}$$
$$\lim_{x \to 0^-} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \infty$$
Más detalles con x→0 a la izquierda$$\lim_{x \to 0^+} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \infty$$
Más detalles con x→0 a la derecha$$\lim_{x \to 1^-} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \log{\left(- \log{\left(\sin{\left(1 \right)} \right)} \right)} - \log{\left(\log{\left(3 \right)} \right)} + i \pi$$
Más detalles con x→1 a la izquierda$$\lim_{x \to 1^+} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \log{\left(- \log{\left(\sin{\left(1 \right)} \right)} \right)} - \log{\left(\log{\left(3 \right)} \right)} + i \pi$$
Más detalles con x→1 a la derecha$$\lim_{x \to -\infty} \log{\left(\frac{\log{\left(\sin{\left(x \right)} \right)}}{\log{\left(3 \right)}} \right)} = \log{\left(\frac{\log{\left(\left\langle -1, 1\right\rangle \right)}}{\log{\left(3 \right)}} \right)}$$
Más detalles con x→-oo