Limiting behavior of the area of an infinity-gon

Calculus Level 3

Suppose a regular \(n\)-gon is constructed from \(N\) sides of constant length \(2\sqrt{\pi}\).

As the number of sides \(N\) of this \(n\)-gon is increased towards infinity, which function of \(N\) gives the limiting behavior of of the \(n\)-gon's area, \(A(N)\)?

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