real analysis - Limit of $n -sqrt{n^2+2n}$

${b_n} = n -\sqrt{n^2+2n}$. Taking $(\frac{1}{n})\rightarrow 0 $ as given, using Algebraic Limit Theorem, show $\lim(b_n)$ exists and find value

The question also says to use the fact that if $(x_n)\rightarrow x$ then $\sqrt{x_n}\rightarrow \sqrt{x}$. I've simplified the ${b_n}$ down to $n(1-\sqrt{1+\frac{2}{n}})$, which appears to be going to zero as n gets large using the algebraic limit theorem. However, I know that $\lim{b_n} = -1$. I'm stuck on how to prove this, and I haven't found any help elsewhere. Any advice would be helpful!