How can I accurately compute $\sqrt{x + 2} −\sqrt{x}$ when $x$ is large? How can the values of the function $f(x) = \sqrt{x + 2} −\sqrt{x}$ be computed accurately when $x$ is large?
I have tried using Matllab. I am not able to understand when $x$ will be large.

In fact, an asymptotic expansion of $f(x) = \sqrt{x+a} - \sqrt{x}$ about $x = \infty$ is $$f(x) \approx \frac{a}{2} x^{-1/2} - \frac{a^2}{8} x^{-3/2} + \frac{a^3}{16} x^{-5/2} - \frac{5a^4}{128} x^{-7/2} + \frac{7a^5}{256} x^{-9/2} - \frac{21a^6}{1024} x^{-11/2} + \cdots.$$  We can calculate this via the generalized binomial theorem $$(a+x)^r = \sum_{k=0}^\infty \binom{r}{k} a^k x^{r-k},$$ for $r = 1/2$, so that in particular, $$\begin{align*} \binom{1/2}{k} &= \frac{(1/2)(1/2-1)\cdot \ldots \cdot(1/2-k+1)}{k!} \\ &= \frac{1(-1)(-3)\cdot \ldots \cdot(3-2k)}{2^k k!} \\ &=  (-1)^{k-1} \frac{(2k-2)!}{2^{2k-1} k! (k-1)!} \\ &= \frac{(-1)^{k-1}}{2^{2k-1}k } \binom{2k-2}{k-1}, \quad k = 1, 2, \ldots,\end{align*}$$ and $\binom{1/2}{0} = 1$.
