Let

\(\displaystyle \alpha = \lim_{m \to \infty} \lim_{n \to \infty } [\cos(n! \pi x)]^{2m}\)

When x is rational

\(\displaystyle \beta = \lim_{m \to \infty }\lim_{n \to \infty } [cos(n! \pi x)]^{2m}\)

When x is irrational

Then the area of the triangle formed by vertices \((\alpha,\beta)\), \((\pi^{e}, -\pi\)), \((e^{\pi}, e\)) is A

Then find 'A + 3'

Also try Crazy Limits!!! 1

**Details**

- answer to nearest integer

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