[tex]l=\lim\limits_{x\to \infty} \dfrac{\arcsin(\sin x)}{x}\\\\ -1\leq\sin x\leq 1 \Big|\arcsin \\ \arcsin(-1)\leq \arcsin(\sin x)\leq \arcsin (1) \\ \\ -\dfrac{\pi}{2}\leq \arcsin(\sin x) \leq \dfrac{\pi}{2}\Big|:x \\\\ -\dfrac{\pi}{2x}\leq \dfrac{\arcsin(\sin x)}{x}\leq \dfrac{\pi}{2x}\\ \\ \lim\limits_{x\to \infty}-\dfrac{\pi}{2x} \leq \lim\limits_{x\to \infty} \dfrac{\arcsin(\sin x)}{x}\leq \lim\limits_{x\to \infty}\dfrac{\pi}{2x}\\ \\ 0\leq \lim\limits_{x\to \infty} \dfrac{\arcsin(\sin x)}{x}\leq 0[/tex]
[tex]\Rightarrow \lim\limits_{x\to \infty} \dfrac{\arcsin(\sin x)}{x} = 0[/tex]
