<s>[INST] 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.
 [/INST] If directly computing $\sqrt{x+2}-\sqrt{x}$ is giving you problems, you can instead try using 
$\sqrt{x+2}-\sqrt{x} = \dfrac{(\sqrt{x+2}-\sqrt{x})(\sqrt{x+2}+\sqrt{x})}{\sqrt{x+2}+\sqrt{x}} = \dfrac{(x+2)-x}{\sqrt{x+2}+\sqrt{x}} = \dfrac{2}{\sqrt{x+2}+\sqrt{x}}$. 
For large $x$, this is approximately $\dfrac{2}{\sqrt{x}+\sqrt{x}} = \dfrac{1}{\sqrt{x}}$.
EDIT: I just tried this for $x = 10^{18}$ using MATLAB R2013b. 
For $\sqrt{x+2}-\sqrt{x}$ it gives $0$, but for $\dfrac{2}{\sqrt{x+2}+\sqrt{x}}$, it gives $1.0000 \times 10^{-9}$. 
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