Question: This is Chapter 3, Exercise 2 of Rudin's Principles.

Calculate $\lim\limits\_{n \to \infty} \sqrt{n^2+n} -n$.

Hints will be appreciated.

Answer: 
When you have a sequence that involves difference between two roots, it's often a good idea to try using the identity $(a-b)(a+b) = a^2-b^2$ to git rid of the root difference:
$$
\sqrt{a} - \sqrt{b} = \left(\sqrt{a} - \sqrt{b}\right)\frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} + \sqrt{b}} = \frac{\left(\sqrt{a} - \sqrt{b}\right)\left(\sqrt{a} + \sqrt{b}\right)}{\sqrt{a} + \sqrt{b}} = \frac{a - b}{\sqrt{a} + \sqrt{b}}
$$
To get an idea of what the answer *might* be, I tried inputting values: n=100, n=1000, n=1000000.
e.g. for n = 1000,
sqrt(1000000 + 1000) ~ 1000 + 1/2 because (1000 + 1/2)^2 = 1000000 + 2\*1000\*(1/2) + 1/4 ~ 1000000 + 1000.

If you can spot this, then the rest is easy.