<s>[INST] What's the trick to get the limit of a sequence with roots like $a_n = -3n + \sqrt{9n^2+11n+16}$? I know of some "tricks" for limits of sequences like
$$b_n = \frac{n^2+7}{3n+1} = \frac{n^2+7}{3n+1} \cdot \frac{\frac{1}{n^2}}{\frac{1}{n^2}} = \frac{\frac{n^2}{n^2}+\frac{7}{n^2}}{\frac{3n}{n^2}+\frac{1}{n^2}}$$
Surely, there has to be a trick like this for the above example $a_n = -3n + \sqrt{9n^2+11n+16}$? Thanks. :-)
 [/INST] \begin{align*}
 -3n + \sqrt{9n^2+11n+16}&=\frac{(-3n + \sqrt{9n^2+11n+16})(-3n - \sqrt{9n^2+11n+16})}{-3n - \sqrt{9n^2+11n+16}}\\
&=\frac{9n^2-(9n^2+11n+16)}{-3n - \sqrt{9n^2+11n+16}}
\end{align*}
</s>