\lim_{n \rightarrow \infty} (\sqrt{n^2+2n}-n) =\lim_{n \rightarrow \infty}\frac{2n}{\sqrt{n^2+2n}+n}=\lim_{n \rightarrow \infty}\frac{2}{\sqrt{1+2/n}+1}=\frac{2}{1+1}=1