A towering limit

Calculus Level 3

\[ \Large \lim_{n\to\infty} \left( n \sin \dfrac \pi n\right)^{\left( 1 + \frac \pi n\right)^n} \]

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

If this limit can be approximated as \(3.19\times 10^\eta\), where \(\eta \) is an integer, find \(\eta\).

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


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