That is, I am looking for an algebraic function $f(x)$ that approximates $\arctan x$ for large values of $x$.
The approximation could be reasonably modest -- perhaps something like
$$\tan (f(x)) = \frac{\pi}{4} + O\left(\frac{1}{x^2}\right).$$
$\endgroup$ 13 Answers
$\begingroup$Use the following
$$\arctan(x) + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} \Leftrightarrow$$
$$\arctan(x) = \frac{\pi}{2}-\arctan\left(\frac{1}{x}\right)$$
The series for $\arctan(x)$ is
$$\arctan(x)=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\ldots$$
Now substitute $\frac1x$ to get the result
$$\arctan(x)=\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^3}-\frac{1}{5x^5} +\ldots$$
So for large values of $x$
$$\arctan(x)\sim\frac{\pi}{2}$$
which is logical, because $\displaystyle \lim_{x\to\infty}\arctan(x)=\frac{\pi}{2}$.
$\endgroup$ $\begingroup$$$ \arctan\left(x\right) ={\pi \over 2}\,{\rm sgn}\left(x\right) - \arctan\left(1 \over x\right) \approx{\pi \over 2}\,{\rm sgn}\left(x\right) - {1 \over x}\,,\qquad \left\vert\,x\,\right\vert \gg 1 $$
$\endgroup$ $\begingroup$This approximation converges much faster:$$ \arctan(x)=2\sum_{n=1}^{\infty}{\frac{1}{2n-1}\frac{{{a}_{n}}\left(x\right)}{a_{n}^{2}\left(x\right)+b_{n}^{2}\left(x\right)}}, $$where$$ \begin{align} & a_1(x)=2/x,\\ & b_1(x)=1,\\ & a_n(x)=a_{n-1}(x)\,\left(1-4/x^2\right)+4b_{n-1}(x)/x,\\ & b_n(x)=b_{n-1}(x)\,\left(1-4/x^2\right)-4a_{n-1}(x)/x. \end{align} $$Therefore, by taking the first term only, the following asymptotic approximation can be used:$$ \arctan\left(x\right) ={\pi \over 2}\,{\rm sgn}\left(x\right) - \arctan\left(1 \over x\right) \approx \frac{\pi}{2}{\rm sgn}(x)-\frac{4 x}{4 x^2+1}\,,\qquad \left\vert\,x\,\right\vert \gg 1. $$
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