lim-x-pi-2-ln-sin-2-x-pi-2-x-2-

Question Number 66316 by mathmax by abdo last updated on 12/Aug/19

{l i m}_{x \to \frac{π}{2}} \frac{l n ({s i n}^{2} x)}{{(\frac{π}{2} - x)}^{2}}

Commented by mathmax by abdo last updated on 21/Aug/19

l e t f (x) = \frac{l n ({s i n}^{2} x)}{{(\frac{π}{2} - x)}^{2}} c h a n g e m e n t \frac{π}{2} - x = t g i v e

{l i m}_{x \to \frac{π}{2}} f (x) = {l i m}_{t \to 0} \frac{l n ({c o s}^{2} t)}{t^{2}} = {l i m}_{t \to 0} \frac{l n (\frac{1 + c o s (2 t)}{2})}{t^{2}}

b u t l n (\frac{1 + c o s (2 t)}{2}) = l n (1 + c o s (2 t)) - l n (2)

c o s (2 t) \sim 1 - 2 t^{2} + o (t^{2}) \Rightarrow 1 + c o s (2 t) \sim 2 - 2 t^{2} \Rightarrow

l n (1 + c o s (2 t)) \sim l n (2) + l n (1 - t^{2}) \sim l n (2) - t^{2} \Rightarrow

\frac{l n (\frac{1 + c o s (2 t)}{2})}{t^{2}} \sim - 1 (t \to o) \Rightarrow {l i m}_{x \to \frac{π}{2}} f (x) = - 1

Answered by kaivan.ahmadi last updated on 12/Aug/19

\overset{h o p}{=} {l i m}_{x \to \frac{π}{2}} \frac{s i n 2 x}{- 2 {s i n}^{2} x (\frac{π}{2} - x)} \overset{h o p}{=}

{l i m}_{x \to \frac{π}{2}} \frac{2 c o s 2 x}{- 2 s i n 2 x (\frac{π}{2} - x) + 2 {s i n}^{2} x} =

\frac{- 2}{2} = - 1