Prove $\lim_\limits{n \to \infty}-\cos(\pi\sqrt{4n^2+10})$ exists and is equal to $-1$ My idea is to take two subsequence converging to different limits. On this step I have problem. How can I set equation  of subsequence when $\cos$ has certain value and $n \to \infty$?
$$
\lim_{n \to \infty}-\cos(\pi\sqrt{4n^2+10})
$$
If I say that $\frac{5}{4n^2} \to 0$ than $n \to \infty$, I have $\lim_{n \to \infty}-\cos(2\pi n) = -1$. Is this right?

HINT:
The cosine is continuous and $$\pi\sqrt{4n^2+10}= 2\pi n\left(1+\frac5{4n^2}+O\left(\frac{1}{n^4}\right)\right)$$
