Aloha :)
a) \(\;\lim\sup\limits_{x\to\infty}\left|\frac{\sin x}{1}\right|=1<\infty\quad\Rightarrow\quad\sin x=O(1)\)
b) \(\;\lim\sup\limits_{x\to\infty}\left|\frac{\tan x}{1}\right|=\lim\sup\limits_{x\to\infty}\left|\frac{\sin x}{\cos x}\right|=\infty\quad\Rightarrow\quad\tan x\ne O(1)\)
c) \(\;\lim\sup\limits_{x\to\infty}\left|\frac{\sin x}{\cos x}\right|=\infty\quad\Rightarrow\quad\sin x\ne O(\cos x)\)
d) \(\;\lim\sup\limits_{n\to\infty}\left|\frac{n!}{n^n}\right|=\lim\sup\limits_{n\to\infty}\left|\frac{1\cdot2\cdots n}{n\cdot n\cdots n}\right|\le1<\infty\quad\Rightarrow\quad n!=O(n^n)\)
