Answer
$$0$$
Work Step by Step
Since
$$a_{n}=\sqrt{n+5}-\sqrt{n+2}$$
Then
\begin{align*}
\lim _{n \rightarrow \infty} a_{n}&=\lim _{n \rightarrow \infty} \sqrt{n+5}-\sqrt{n+2} \\
&= \lim _{n \rightarrow \infty} (\sqrt{n+5}-\sqrt{n+2})\frac{(\sqrt{n+5}+\sqrt{n+2})}{\sqrt{n+5}+\sqrt{n+2}}\\
&=\lim _{n \rightarrow \infty}\frac{(n+5)-(n+2)}{\sqrt{n+5}+\sqrt{n+2}}=\frac{3}{\sqrt{n+5}+\sqrt{n+2}} \\
&=\lim _{n \rightarrow \infty} \frac{3}{\sqrt{n+5}+\sqrt{n+2}}\\
&=0
\end{align*}
