Answer
$\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent.
Work Step by Step
Consider $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty}
\dfrac{n+3}{n^2+5n+6}$
$= \lim\limits_{n \to \infty} \dfrac{n+3}{(n+3)(n+2)}$
Thus, $\lim\limits_{n \to \infty} \dfrac{1}{n+2}=0$
Thus, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent.
