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How can be explained, in a "clean" mathematical way, the "stabilization" of the different digits in the following divergent sequence?

$$\tan(89°)=57.28996163...$$

$$\tan(89.9°)=572.9572134...$$

$$\tan(89.99°)=5729.577893...$$

$$\tan(89.999°)=57295.77951...$$

$$\tan(89.9999°)=572957.7951...$$

etc...

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  • $\begingroup$ I don't know but I guess it has something to do with the fact that $$\lim_{n\to\infty}\frac{\cot10^{-n-1}}{\cot10^{-n}}=10\ ?$$ $\endgroup$ – bof Oct 20 '16 at 11:05
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    $\begingroup$ What is paradoxical? $\endgroup$ – mrf Oct 20 '16 at 11:35
  • $\begingroup$ @mrf I mean that there is an apparent paradox in having a convergence of the digits whereas the sequence is divergent... $\endgroup$ – Jean Marie Oct 20 '16 at 11:39
  • $\begingroup$ @bof You are right, due to the series for $\cot(x)=1/x+...$, but why these digits $5,7,2,9,5,7...$ $\endgroup$ – Jean Marie Oct 20 '16 at 11:43
  • $\begingroup$ Did you check the suggested duplicate? $180/\pi = 57.2957795131\ldots$. $\endgroup$ – mrf Oct 20 '16 at 11:46
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$\frac{\sin{x}}{x} \to 1$ when $x \to 0$, so $\cos(90^\circ-x)=\sin{x} \approx x$, meaning when you're $10$ times closer to $90^\circ$ cosine is $10$ times smaller and the tangent is $10$ times bigger because sine is approximately equal to one.

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