A towering limit

Calculus Level 2

limn(nsinπn)(1+πn)n \Large \lim_{n\to\infty} \left( n \sin \dfrac \pi n\right)^{\left( 1 + \frac \pi n\right)^n}

The limit above has a closed form. Find the value of this closed form.

If this limit can be approximated as 3.19×10η3.19\times 10^\eta, where η\eta is an integer, find η\eta.

You may use a calculator for the final step of your calculation.

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