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: 
$\sqrt{n^2+n} - n= \frac{n}{\sqrt{n^2+n} + n} = \frac{1}{\sqrt{1+\frac 1n} + 1}$
You can also do this by inductively showing it is increasing and bounded, but their ideas are probably a better route.