I'd love your help with finding the following limit:
$$\lim_{n\to \infty }\cos (\pi\sqrt{n^{2}-n}).$$

I was asked to find this limit, but honestly I believe that it doesn't exist.

According to Heine Theorem of limit of functions, I can choose two sequences:

$x_{k}=2\pi k$ and $y_{k}=2\pi k+\pi$ and notice that when I apply the function on both of them, I'll get -1 and 1, respectively.

Am I right?

Thank you again.

We have 
\begin{align*}
\cos (\pi\sqrt{n^2-n})&= (-1)^n\cos(\pi(\sqrt{n^2-n}-n))\\
&= (-1)^n \cos\pi\frac{-n}{\sqrt{n^2-n}+n}\\
&=(-1)^n\cos \pi\frac 1{\sqrt{1-\frac 1n }+1},
\end{align*}
hence $|\cos(\pi\sqrt{n^2-n})| = \left|\cos \left(\pi\frac 1{\sqrt{1-\frac 1n }+1}\right)\right|$. 
Since $\lim \limits_{n\to +\infty}\pi\frac 1{\sqrt{1-\frac 1n }+1} =\frac{\pi}2$, the $\cos$ is continuous and $\cos \frac{\pi}2 =0$ we conclude that the limit is $0$.