lim-x-pi-4-sec-2-x-2-tan-x-2cos-x-2sin-x-

Question Number 190824 by cortano12 last updated on 12/Apr/23

\lim_{x \to \frac{π}{4}} \frac{\sec^{2} x - 2 \sqrt{\tan x}}{2 \cos x - 2 \sin x} = ?

Commented by 0670322918 last updated on 12/Apr/23

\frac{1}{2} \lim_{x \to \frac{π}{4}} \frac{{t a n}^{2} (x) + 1 - 2 + 2 - 2 \sqrt{t a n (x)}}{c o s (x) [1 - t a n (x)]} =

c o s (\frac{π}{4}) = \frac{\sqrt{2}}{2}

\frac{1}{\sqrt{2}} \lim_{x \to \frac{π}{4}} [\frac{{t a n}^{2} (x) - 1}{1 - t a n (x)} + 2 \frac{1 - \sqrt{t a n (x)}}{1 - t a n (x)}] =

\frac{1}{\sqrt{2}} [\lim_{x \to + \frac{π}{4}} [2 \frac{1 - \sqrt{t a n (x)}}{[1 - \sqrt{t a n (x)}] [1 + \sqrt{t a n (x)}} - \frac{[t a n (x) - 1] [t a n (x) + 1]}{t a n (x) - 1}] =

\frac{1}{\sqrt{2}} (\lim_{x \to \frac{π}{4}} [\frac{2}{1 + \sqrt{t a n (x)}} - t a n (x) - 1]) = \frac{1}{\sqrt{2}} [1 - 1 - 1] = - \frac{\sqrt{2}}{2}

\lim_{x \to \frac{π}{4}} \frac{{s e c}^{2} (x) - 2 \sqrt{t a n (x)}}{2 c o s (x) - 2 s i n (x)} = - \frac{\sqrt{2}}{2}

Commented by cortano12 last updated on 13/Apr/23

ans \frac{1}{\sqrt{2}}