<s>[INST] Computing limits which involve square roots, such as $\sqrt{n^2+n}-n$ Is there a general strategy for this? For example I'm working on the limit
$$\lim_{n\rightarrow\infty}\sqrt{n^2 + n} - n $$  
I have a simple argument to show that this limit is less than or equal to 1/2, but I can't get much further because it's difficult for me to manipulate the square root symbol. Here is the argument:
$\sqrt{n^2 + n} - n \lt \sqrt{n^2 +n + \frac{1}{4}} - n = n+\frac{1}{2} - n = 1/2$
 [/INST] $\begin{align}
\sqrt{n^2 + n} - n & = (\sqrt{n^2 + n} - n) \cdot \frac{\sqrt{n^2 + n} + n}{\sqrt{n^2 + n} +n} \\
& = \frac{n^2+n-n^2}{\sqrt{n^2 + n} + n} \\
& = \frac{1}{\frac{\sqrt{n^2 + n} + n}{n}} \\
& = \frac{1}{\sqrt{1 + \frac{1}{n}} + 1} \rightarrow \frac{1}{2}\\
\end{align}$
</s>