text: ## Thomas' Calculus 13th Edition
  
  $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent.
  Let $\lim\limits_{n \to \infty} a_n= \lim\limits_{n \to \infty} \dfrac{n+3}{n^2+5n+6}$ or, $\lim\limits_{n \to \infty} \dfrac{n+3}{n^2+5n+6}= \lim\limits_{n \to \infty} \dfrac{n+3}{(n+3)(n+2)}$ Now, apply Sandwich Theorem which states that $\lim\limits_{n \to 0} \dfrac{1}{n}=0$ Thus, $\lim\limits_{n \to \infty} \dfrac{1}{n+2}=0$ Hence, $\lim\limits_{n \to \infty} a_n=0$ and {$a_n$} is convergent.