Trigonometric Limit!

Calculus Level 2

Evaluate

lim ⁡ x → 99 9 + tan ⁡ − 1 ( 1 x − 999 ) . \displaystyle \lim_{x \to 999^+} \tan^{-1} \left(\frac{1}{x - 999}\right).

Details and assumptions

tan ⁡ − 1 x \tan^{-1}x denotes the inverse of tan ⁡ x \tan x and not the reciprocal 1 tan ⁡ x \frac{1}{\tan x} .

The principal branch of tan ⁡ − 1 \tan^{-1} is ( − π 2 , π 2 ) \left( -\frac{\pi}{2}, \frac{\pi}{2}\right) .


The answer is 1.5707963268.

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Calvin Lin by Calvin Lin

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1 solution

Ricky Escobar
Dec 17, 2013

Let u = 1 x − 999 u=\frac{1}{x-999} , then as x → 99 9 + , u → ∞ x \to 999^+, \ u \to \infty . Then our limit is lim ⁡ u → ∞ tan ⁡ − 1 u = π 2 ≈ 1.571 . \lim_{u \to \infty} \tan^{-1} u = \frac{\pi}{2} \approx \boxed{1.571}.

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