Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

What value of \(N\) satisfies

\[ \lim_{n \rightarrow \infty} \left( 1 + \frac{5} { 2n} \right) ^ {10n} = e^{N} ? \]

What is the value of the limit \[\lim_{n \to \infty} \frac{1}{n} \sin \frac{n}{7}\pi?\]

What is \(\displaystyle \lim_{n \to \infty} 2(\sqrt{n+6}\sqrt{n+10}-n) \)?

Evaluate \[\lim_{n\to\infty}\sin\frac{n\pi}{2}.\]

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