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Evaluate
limx→999+tan−1(1x−999). \displaystyle \lim_{x \to 999^+} \tan^{-1} \left(\frac{1}{x - 999}\right). x→999+limtan−1(x−9991).
Details and assumptions
tan−1x\tan^{-1}xtan−1x denotes the inverse of tanx\tan xtanx and not the reciprocal 1tanx\frac{1}{\tan x}tanx1.
The principal branch of tan−1 \tan^{-1}tan−1 is (−π2,π2)\left( -\frac{\pi}{2}, \frac{\pi}{2}\right)(−2π,2π).
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