Evaluate the limit of a subsequence: Compute the limit $ \lim_{n\to\infty} \cos\left( \pi \sqrt{4n^2 + 5n + 1} \right) $ Integer $n$ on Mathematica


I want to compute the limit $ $ \lim \limits_{n\to\infty} \cos\left( \pi \sqrt{4n^2 + 5n + 1} \right) $ $ for integer $ n$ . By completing the square, we can determine that this limit is equal to $ – \tfrac1{\sqrt2} \approx -0.7071 $ .

But if we don’t restrict $ n$ to an integer, then the limit is indeterminate / does not exist. And can be easily found by typing it on WolframAlpha. Or in Mathematica:

However, I do not know how to compute the limit (on Mathematica) with the original constraint that $ n$ must be an integer.

I know that we can plot a graph on Mathematica:

But what is the exact value of the limit?

The graph suggests that the limit is equal to $ -\tfrac1{\sqrt2} $ . However, this doesn’t look like a convincing result because we can’t know that the limit is exactly equal to $ -\tfrac1{\sqrt2} $ .

Question: Is there a way to compute this limit in Mathematica where it spits out a single numerical value (of $ -1/{\sqrt2}$ )?