\lim_{n\to \infty } \, \frac{n \sin (n!)}{n^2}=\lim_{n\to \infty } \, \frac{ \sin (n!)}{n}=\lim_{n\to \infty } \frac{1}{n}\lim_{n\to \infty } \sin (n!)=0*\{-1,1\}=0