<s>[INST] How to approximate a function involving arc-tangent for a sufficiently large argument? I have a function, $f(x) = {\tan ^{ - 1}}(\sqrt {1 + {x^2}}  - x)$. Let's assume the argument $x$ is sufficiently high. If so, how can I approximate $f(x)$ in terms of $x$?  I have computed $f(x)$ with MATLAB. It looks like that $f(x)$ can be approximated to ${1 \over {2x}}$. That is, ${\tan ^{ - 1}}(\sqrt {1 + {x^2}}  - x) \approx {1 \over {2x}}$ for a large $x$. It this right?
 [/INST] As Martin Argerami already answered, you are totally write.
If you want more accuracy, you could use Taylor series
$$\sqrt{1+x^2}=x+\frac{1}{2 x}-\frac{1}{8 x^3}+\frac{1}{16
   x^5}+O\left(\frac{1}{x^7}\right)$$
$$\sqrt{1+x^2}-x=\frac{1}{2 x}-\frac{1}{8 x^3}+\frac{1}{16
   x^5}+O\left(\frac{1}{x^7}\right)$$ Now, using
$$\tan^1(\epsilon)=\epsilon -\frac{\epsilon ^3}{3}+\frac{\epsilon ^5}{5}+O\left(\epsilon ^7\right)$$ you should arrive to 
$$f(x) = {\tan ^{ - 1}}(\sqrt {1 + {x^2}}  - x)=\frac{1}{2 x}-\frac{1}{6 x^3}+\frac{1}{10
   x^5}+O\left(\frac{1}{x^7}\right)$$ or more generally
$$f(x) = {\tan ^{ - 1}}(\sqrt {1 + {x^2}}  - x)=\sum_{i=1}^\infty \frac{(-1)^{i-1} }{(4 i-2)x^{2 i-1}}$$
Using $x=5$ (which is quite small), the truncated series given above leads to $\frac{9253}{93750}\approx 0.0986987$ while the "exact" value would be $\approx 0.0986978$. 
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