lim as n goes to infinity of square root of (n + square root of n+1) - square root n)
does it equal 0
...and...? where are you getting stuck?
just to make things a little less messy, note that if you make a substitution of $\displaystyle t = \sqrt{n + 1}$ then your limit is the same as
$\displaystyle \lim_{t \to \infty} \frac t{\sqrt{t^2 + t - 1} + \sqrt{t^2 - 1}}$
now can you finish up?