Precise asymptotics of the sum of two square root functions

Question

Question: Let $x_n$ be some term that goes to infinity as $n\rightarrow\infty$. I would like to know the precise asymptotics of $-\sqrt{x_n}+\sqrt{x_n+4}$.

My attempt: We have \begin{align*} -\sqrt{x_n}+\sqrt{x_n+4} &=\sqrt{x_n}\left(-1+\sqrt{1+\frac{4}{x_n}}\right) \\ &=\sqrt{x_n}(-1+1+\epsilon_n) \\ &=\sqrt{x_n}\cdot\epsilon_n, \end{align*} where $\epsilon_n$ is some decaying function that I want the precise asymptotics of. Thanks.

Answer 1

$$\sqrt{x_n+4}-\sqrt{x_n}=\frac{4 }{\sqrt{x_n+4}+\sqrt{x_n}} \sim \frac{2 }{\sqrt{x_n}}$$

Answer 2

Your start is fine, now using the Taylor series $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^2}{8}+\cdots$$ you obtain $$\sqrt{x_n}\left(-1+1+\frac{2}{x_n}-\frac{2}{x_n^2}+\dots\right)=\frac{2}{x_n^{1/2}}-\frac{2}{x_n^{3/2}}+\dots\overset{x_n\to\infty}{\longrightarrow} 0$$ as the series is alternating and monotonically decreasing