¿Cómo puedo encontrar a $\lim \sup\text{ or } \lim \inf$$ \sin (\frac{n\pi}{5})$ ?
Sé el $\sin$ función oscila normalmente entre el$-1$$1$, pero que obviamente no es la respuesta para $\lim \inf$$\lim \sup$.
Respuesta
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Observar que $$\begin{align*}&\left\{ \sin \left(\frac{n\pi}{5}\right)|n\in \mathbb{N}\right\}\\&=\left\{\sin \left(\frac{\pi}{5}\right),\sin \left(\frac{2\pi}{5}\right),\sin \left(\frac{3\pi}{5}\right),\sin \left(\frac{4\pi}{5}\right),\sin \left(\frac{5\pi}{5}\right),\sin \left(\frac{6\pi}{5}\right),\sin \left(\frac{7\pi}{5}\right),\sin \left(\frac{8\pi}{5}\right),\sin \left(\frac{9\pi}{5}\right),\sin \left(\frac{10\pi}{5}\right)\right\} \\&=\left\{0,\sin \left(\frac{\pi}{5}\right),\sin \left(\frac{2\pi}{5}\right),\sin \left(\frac{3\pi}{5}\right),\sin \left(\frac{4\pi}{5}\right),-\sin \left(\frac{\pi}{5}\right),-\sin \left(\frac{2\pi}{5}\right),-\sin \left(\frac{3\pi}{5}\right),-\sin \left(\frac{4\pi}{5}\right)\right\}.\end{align*}$$
Por lo tanto
$$\begin{align*}&\limsup_{n\to\infty} \sin \left(\frac{n\pi}{5}\right)\\&=\max \left\{0,\sin \left(\frac{\pi}{5}\right),\sin \left(\frac{2\pi}{5}\right),\sin \left(\frac{3\pi}{5}\right),\sin \left(\frac{4\pi}{5}\right),-\sin \left(\frac{\pi}{5}\right),-\sin \left(\frac{2\pi}{5}\right),-\sin \left(\frac{3\pi}{5}\right),-\sin \left(\frac{4\pi}{5}\right)\right\}\end{align*} $$ y
$$\begin{align*}&\liminf_{n\to\infty} \sin \left(\frac{n\pi}{5}\right)\\&=\min \left\{0,\sin \left(\frac{\pi}{5}\right),\sin \left(\frac{2\pi}{5}\right),\sin \left(\frac{3\pi}{5}\right),\sin \left(\frac{4\pi}{5}\right),-\sin \left(\frac{\pi}{5}\right),-\sin \left(\frac{2\pi}{5}\right),-\sin \left(\frac{3\pi}{5}\right),-\sin \left(\frac{4\pi}{5}\right)\right\}.\end{align*} $$
