Answer
$0$
Work Step by Step
We have
$$ \lim\limits_{n \to \infty}{d_n}=\lim\limits_{n \to \infty} \sqrt{n+3}-\sqrt{n}\\
=\lim\limits_{n \to \infty} (\sqrt{n+3}-\sqrt{n} )\frac{\sqrt{n+3}+\sqrt{n}}{\sqrt{n+3}+\sqrt{n}}\\
=\lim\limits_{n \to \infty} \frac{ 3}{\sqrt{n+3}+\sqrt{n}}\\
=\lim\limits_{n \to \infty} \frac{ 3/\sqrt{n}}{\sqrt{1+3/n}+1}=0.$$
Using Theorem 1, we see that the sequence $a_n$ converges to $0$.
