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I want to compute the limit

$\qquad \lim \limits_{n\to\infty} \cos\left( \pi \sqrt{4n^2 + 5n + 1} \right)$

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 $-1/{\sqrt2}$)?

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  • $\begingroup$ Actually, can you show the analytical derivation of the result? It should help also coming up with how to coerce Mathematica into solving the limit. $\endgroup$ – Kiro Oct 9 '20 at 10:11
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    $\begingroup$ Hi Kiro. Karen has posted an analytical solution here. If you're too lazy to sign in to see the solution, just see this image. $\endgroup$ – GohP.iHan Oct 9 '20 at 11:19
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    $\begingroup$ Thank you (user64494, Vaclav, and Kiro) for your swift response. This is the first time I'm posting on Mathematica Stackexchange, and I'm happy to see such an active and helpful community! $\endgroup$ – GohP.iHan Oct 9 '20 at 11:28
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Improving previous answers,

DiscreteLimit[Normal[Series[Cos[Pi*Sqrt[4 n^2 + 5 n + 1]],{n,Infinity,1}]],n -> Infinity]
(*-(1/Sqrt[2])*)
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  • $\begingroup$ This is what I'm looking for. In fact, I was hoping to generalize my question where the domain of $n$ is of my own choosing. For example, if $n$ is half or any integer, or if $n$ can only be a prime number, then the limit remains the same. Let me try to tweak your code to see how to get this done. On the other hand, feel free to leave some hints here to steer me in the right direction. $\endgroup$ – GohP.iHan Oct 9 '20 at 11:25
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DiscreteAsymptotic[Cos[π*
   Sqrt[4*n^2 + 5 n + 1]], n -> Infinity](*-(1/Sqrt[2])*)
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  • $\begingroup$ Thank you too, A little mouse on the pampas. $\endgroup$ – GohP.iHan Oct 9 '20 at 11:46
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    $\begingroup$ +1.Moreover, DiscreteAsymptotic[Cos[\[Pi]*Sqrt[4*n^2 + 5 n + 1]], n -> Infinity, SeriesTermGoal -> 2] produces three terms $$\frac{\frac{81 \pi ^2}{8192 \sqrt{2}}+\frac{45 \pi }{512 \sqrt{2}}}{n^2}-\frac{9 \pi }{64 \sqrt{2} n}-\frac{1}{\sqrt{2}} ,$$ not two ones (despite the documentation) as a bonus. $\endgroup$ – user64494 Oct 9 '20 at 15:27
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We have

 Normal[Series[Sqrt[4*n^2 + 5 n + 1], {n, Infinity, 1}]]
 (* 5/4 - 9/(64 n) + 2 n *)

If n tends to infinity, the expression above tends to

 Cos[(5/4 + 2 n) \[Pi]]

For integer n is the limit equal to

$\cos \left(\frac{5 \pi }{4}\right)=-\frac{1}{\sqrt{2}}$

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  • $\begingroup$ Thank you for your solution. I'm reluctant to edit the expression of the limit because I wanted to have the ability to know how to directly compute limits like this (for integer $n$) on Mathematica without manipulating the expression. $\endgroup$ – GohP.iHan Oct 9 '20 at 11:21
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A bit of a patchwork way to show that the limit is indeed -1/Sqrt[2] for integer values of n goes as follows.

We make a series expansion of Sqrt[4n^2+5n+1] at infinity.

ser = Series[Sqrt[4 n^2 + 5 n + 1], {n, \[Infinity], 3}];

Then by keeping in mind that Cos is periodic with a period of 2 Pi, we can do

Cos[Limit[ser - 2 n, n -> \[Infinity]] Pi]
(*-(1/Sqrt[2])*)

where ser-2n is effectively taking the result modulo 2Pi and is valid if and only if n is an integer.

I admit that this is a bit sketchy and unsatisfactory way of doing this. While we can fully justify what we do, if we made a mistake in our thinking then the result is also wrong, so correctness of the result depends significantly on things not taken care of by Mathematica. I expect there to be more elegant ways.

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  • $\begingroup$ I did not consider converting the expression into a series in Mathematica at all. Thank you for showing me an alternative approach. $\endgroup$ – GohP.iHan Oct 9 '20 at 11:27

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