lim-x-pi-2-sec-2-x-sec-x-tan-x-

Question Number 123375 by bemath last updated on 25/Nov/20

\lim_{x \to π / 2} (\sec^{2} x - \sec x \tan x) = ?

Answered by Dwaipayan Shikari last updated on 25/Nov/20

\lim_{x \to \frac{π}{2}} (\frac{1 - s i n x}{{c o s}^{2} x}) = \frac{2 ({s i n}^{2} (\frac{π}{4} - \frac{x}{2}))}{{s i n}^{2} (\frac{π}{2} - x)} = \frac{2 {(\frac{π}{4} - \frac{x}{2})}^{2}}{{(\frac{π}{2} - x)}^{2}} = 2 . \frac{1}{4} = \frac{1}{2}

Answered by liberty last updated on 25/Nov/20

\underset{x \to π / 2}{\lim s e c} x (\sec x - \tan x) =

\lim_{x \to π / 2} \frac{(\frac{1 - \sin x}{\cos x})}{\cos x} = \lim_{x \to π / 2} \frac{1 - \sin x}{\cos^{2} x}

[l e t x = \frac{π}{2} + y; y \to 0]

\lim_{y \to 0} \frac{1 - \cos y}{\sin^{2} y} = \lim_{y \to 0} \frac{2 \sin^{2} (y / 2)}{\sin^{2} y} = \frac{1}{2}

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