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: 
$\sqrt{n^2+n} - n= \frac{n}{\sqrt{n^2+n} + n} = \frac{1}{\sqrt{1+\frac 1n} + 1}$
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.