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Solve the limit of the following sequence?

Posted 3 months ago
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I'd like to solve the Limit of the following Sequence on Mathematica 12 :

DiscreteLimit[(1 + Sin[Pi*Sqrt[4*n^2 + 1]])^n, n -> Infinity]

No results given by Mathematica. But I do some corresponding equivalent deformation, Sin[Pi*Sqrt[4*n^2 + 1]=Sin[-2*n*Pi+Pi*Sqrt[4*n^2 + 1] =Sin[Pi/(Sqrt[4*n^2 + 1] + 2*n)]

DiscreteLimit[(1 + Sin[Pi/(Sqrt[4*n^2 + 1] + 2*n)])^n, n -> Infinity]

I got the corresponding results fortunately: E^([Pi]/4).

What is the effect of the form of the expression on seeking the limit? Which direction should I try to get the result? I want to know some reason for this.

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