Evalute $\lim\limits_{n\to\infty}\sin^2(\pi\sqrt{ n^2+n})$. I like to  find the following limit:

$\lim\limits_{n\to\infty}\sin^2(\pi\sqrt{ n^2+n})$.

Any ideas or insight would be greatly appreciated.

As $n$ tends to $+\infty$, you may write
$$
\begin{align}
\sin^2 \left( \pi \sqrt{n^2+n}\right) &=\sin^2 \left( \pi n \:\sqrt{1+\frac1n}\right)\\\\
&=\sin^2 \left( \pi n \:\left(1+\frac1{2n}+\mathcal{O}\left(\frac{1}{n^2}\right)\right)\right)\\\\
&=\sin^2 \left( \pi n +\frac{\pi}2+\mathcal{O}\left(\frac{1}{n}\right)\right)\\\\
&=\cos^2 \left(\mathcal{O}\left(\frac{1}{n}\right)\right)\to 1.
\end{align}
$$
