1. **Problem:** Find the limit of the sequence $$a_n = \frac{2n^n}{(n+1)^n}$$ as $n \to \infty$. 2. **Formula and rules:** To find limits of sequences involving powers, rewrite expressions to compare growth rates. Use the fact that $$\left(\frac{n}{n+1}\right)^n = \left(1 - \frac{1}{n+1}\right)^n \to e^{-1}$$ as $n \to \infty$. 3. **Work:** $$a_n = \frac{2n^n}{(n+1)^n} = 2 \left(\frac{n}{n+1}\right)^n = 2 \left(1 - \frac{1}{n+1}\right)^n$$ As $n \to \infty$, $$\left(1 - \frac{1}{n+1}\right)^n \to e^{-1}$$ 4. **Conclusion:** $$\lim_{n \to \infty} a_n = 2 e^{-1} = \frac{2}{e}$$ --- Since the user asked for multiple sequences but per instructions we solve only the first, we count all three sequences as distinct problems but solve only the first.