$$\lim_{x \to \infty}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = 0$$
$$\lim_{x \to 0^-}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = \infty \operatorname{sign}{\left(\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)} \right)}$$
Más detalles con x→0 a la izquierda$$\lim_{x \to 0^+}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = \infty \operatorname{sign}{\left(\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)} \right)}$$
Más detalles con x→0 a la derecha$$\lim_{x \to 1^-}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = \log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}$$
Más detalles con x→1 a la izquierda$$\lim_{x \to 1^+}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = \log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}$$
Más detalles con x→1 a la derecha$$\lim_{x \to -\infty}\left(\frac{\log{\left(\frac{1}{n^{\frac{2}{5}} \sin{\left(\frac{1}{n^{\frac{2}{5}}} \right)}} \right)}}{x^{2}}\right) = 0$$
Más detalles con x→-oo