TITLE: Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$
QUESTION [29 upvotes]: 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.

REPLY [6 votes]: Considering the form $\cos(\pi n \sqrt{1-\frac{1}{n}})$ and using
Taylor's expansion for $\sqrt{1-\frac{1}{x}}$ $\to$ see here,
we get that when n is large $\cos (\pi\sqrt{n^{2}-n}) \approx \cos( \pi n -\frac{\pi}{2})$. Therefore, $L=0$.
Q.E.D.