limit of $\sqrt{n+k}-\sqrt{n}$ without Taylor series

Question

Let $a_n = \sqrt{n+k}-\sqrt{n}$ be a sequence where $k\in \Bbb R^+$. Find the limit of $a_n$.

I want to evaluate this limit without doing a series expansion of the terms (it seems like overkill). Intuitively, it should be $0$ because as $n$ becomes large the $k$ becomes insignificant. But how can I show it? I can't just factor out $\sqrt{n}$ because $$\sqrt{n}(\sqrt{1+\frac{k}{n}}-1)$$ goes to $\infty\cdot 0$.

Answer 1

Note that $(\sqrt{n+k}-\sqrt{n})(\sqrt{n+k}+\sqrt{n})=(n+k)-n=k$.

Hence we have that $\sqrt{n+k}-\sqrt{n}= \frac{(\sqrt{n+k}-\sqrt{n})(\sqrt{n+k}+\sqrt{n})}{(\sqrt{n+k}+\sqrt{n})}=\frac{k}{\sqrt{n+k}+\sqrt{n}},$ which goes to zero since the numerator is constant and the denominator goes to infty.