Prove $\frac{2n+\sin(n)}{n+2}$ converges to 2.

Good job!

If you don't have to prove from definition:

\begin{align} \lim_{n \to \infty} \frac{2n + \sin n}{n+2} = \lim_{n \to \infty } \frac{2+ \frac{\sin(n)}{n}}{1+\frac{2}{n}}=2 \end{align}

Alternatively, $$ \frac{2n-1}{n+2} \le \frac{2n+\sin(n)}{n+2} \le \frac{2n+1}{n+2} $$ and both bounds go to $2$ as $n \to \infty$.