Three limits by the squeeze theorem

1. Each term satisfies $\frac n{n^2+n}\le\frac n{n^2+k}\le\frac n{n^2+1}$. Summing the $n$ terms:
$$\frac{n^2}{n^2+n}\le A_n\le\frac{n^2}{n^2+1}.$$
Both bounds tend to $1$, so $A_n\to1$.

2. $3^n\le1+2^n+3^n\le3\cdot3^n$, so taking $n$-th roots, $3\le B_n\le3\cdot3^{1/n}$, and $3^{1/n}\to1$. So $B_n\to3$.

3. Let $M=\max a_i$. Then $M^n\le a_1^n+\dots+a_p^n\le p\,M^n$, so $M\le C_n\le p^{1/n}M$ and $p^{1/n}\to1$. Hence $C_n\to M$.

Remark. The result in 3 says that a sum of powers behaves, to the $n$-th root, like its largest term: the same idea underlies the $\ell^p$ norms tending to the maximum norm as $p\to\infty$.