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: 
Hint:
$$\frac{\sqrt{n^2+n}-n}{1} = \frac{\sqrt{n^2+n}-\sqrt{n^2}}{1}\times \frac{\sqrt{n^2+n}+\sqrt{n^2}}{\sqrt{n^2+n}+\sqrt{n^2}} = \cdots$$
I will expand more if needed.
$\sqrt{n^2+n} - n= \frac{n}{\sqrt{n^2+n} + n} = \frac{1}{\sqrt{1+\frac 1n} + 1}$