5.3. Find the limit of the sequence:

$$
\lim _{n \rightarrow \infty}(\sqrt{(n+2)(n+7)}-n)
$$
Solution. Multiply and divide the general term of the sequence by $\sqrt{(n+2)(n+7)}+n$ and use the formula for the difference of squares:

$$
\begin{gathered}
\lim _{n \rightarrow \infty}\left(\frac{(\sqrt{(n+2)(n+7)}-n) \cdot(\sqrt{(n+2)(n+7)}+n)}{(\sqrt{(n+2)(n+7)}+n)}\right)= \\
=\lim _{n \rightarrow \infty} \frac{\left((n+2)(n+7)-n^{2}\right)}{n\left(\sqrt{((n+2)(n+7)) / n^{2}}+1\right)}=\lim _{n \rightarrow \infty} \frac{9 n+14}{n(\sqrt{(1+2 / n)(1+7 / n)}+1)}= \\
=\lim _{n \rightarrow \infty} \frac{9+14 / n}{(\sqrt{(1+2 / n)(1+7 / n)}+1)}=9 / 2
\end{gathered}
$$