lim-x-pi-2-2-1-sin-x-2-cos-2-x-

Question Number 138996 by bramlexs22 last updated on 21/Apr/21

\lim_{x \to π / 2} \frac{\sqrt{2} - \sqrt{1 + \sin x}}{\sqrt{2} \cos^{2} x} = ?

Answered by EDWIN88 last updated on 21/Apr/21

\lim_{x \to π / 2} \frac{1 - \sin x}{\sqrt{2} (1 - \sin x)} . \lim_{x \to π / 2} \frac{1}{(1 + \sin x) (\sqrt{2} + \sqrt{1 + \sin x})}

= \frac{1}{\sqrt{2}} . \frac{1}{2 (2 \sqrt{2})} = \frac{1}{8}

Answered by mathmax by abdo last updated on 22/Apr/21

let f (x) = \frac{\sqrt{2} - \sqrt{1 + sinx}}{\sqrt{2} \cos^{2} x} \Rightarrow f (x) =_{x = \frac{π}{2} - t} \frac{\sqrt{2} - \sqrt{1 + cost}}{\sqrt{2} \sin^{2} t} = g (t)

(x \to \frac{π}{2} \Rightarrow t \to 0) cost \sim 1 - \frac{t^{2}}{2} \Rightarrow 1 + cost \sim 2 - \frac{t^{2}}{2} and sint \sim t

\sqrt{1 + cost} \sim \sqrt{2 - \frac{t^{2}}{2}} = \sqrt{2} \sqrt{1 - \frac{t^{2}}{4}} \sim \sqrt{2} (1 - \frac{t^{2}}{8}) \Rightarrow

g (t) \sim \frac{\sqrt{2} - \sqrt{2} (1 - \frac{t^{2}}{8})}{\sqrt{2} t^{2}} = \frac{1}{8} \Rightarrow \lim_{t \to 0} g (t) = \frac{1}{8} = \lim_{x \to \frac{π}{2}} f (x)