Interesting Limit (2)

Note that $\begin{aligned} \large\pi n!e&=\large\pi n!\left[2+\frac{1}{2!}+···+\frac{1}{n!}+\frac{1}{(n+1)!}+\omicron\left(\frac{1}{(n+1)!}\right)\right]\\ &=\large\pi a_n+\frac{\pi}{n+1}+\omicron\left(\frac{1}{n+1}\right), \end{aligned}$ where $a_n$ is integer. Hence $\begin{aligned} \large\lim\limits_{n\rightarrow\infty}\frac{n^2\sin^2(\pi n!e)}{\pi^2}&=\large\lim\limits_{n\rightarrow\infty}\frac{n^2}{\pi^2}\sin^2\left[\frac{\pi}{n+1}+\omicron\left(\frac{1}{n+1}\right)\right]\\ &=\large\lim\limits_{n\rightarrow\infty}\frac{n^2}{(n+1)^2}\left[\frac{\sin\left(\frac{\pi}{n+1}+\omicron\left(\frac{1}{n+1}\right)\right)}{\frac{\pi}{n+1}}\right]^2\\ &=\large1. \end{aligned}$