lim-x-pi-2-1-sin-x-x-2-cot-2-x-

Question Number 106173 by john santu last updated on 03/Aug/20

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

Answered by bemath last updated on 03/Aug/20

set x = \frac{π}{2} + ♭ \to \frac{4}{π^{2}} \times \lim_{♭ \to 0} \frac{1 - \sin (\frac{π}{2} + ♭)}{\cot^{2} (\frac{π}{2} + ♭)} =

\frac{4}{π^{2}} \times \lim_{♭ \to 0} \frac{1 - \cos ♭}{\tan^{2} ♭} = \frac{4}{π^{2}} \times \lim_{♭ \to 0} \frac{2 \sin^{2} (\frac{♭}{2})}{\tan^{2} ♭}

= \frac{8}{π^{2}} \times {[\lim_{♭ \to 0} \frac{\sin (\frac{♭}{2})}{\tan ♭}]}^{2}

= \frac{8}{π^{2}} \times \frac{1}{4} = \frac{2}{π^{2}} ★

Answered by mathmax by abdo last updated on 03/Aug/20

let g (x) = \frac{1 - sinx}{x^{2} {cotan}^{2} x} \Rightarrow g (x) = \frac{\sin^{2} x (1 - sinx)}{x^{2} \cos^{2} x}

changement x = \frac{π}{2} - t give g (x) = h (t) = \frac{\cos^{2} t (1 - cost)}{{(\frac{π}{2} - t)}^{2} \sin^{2} t}

(x \to \frac{π}{2} \Rightarrow t \to 0) \Rightarrow h (t) \sim \frac{{(1 - \frac{t^{2}}{2})}^{2} . \frac{t^{2}}{2}}{{(\frac{π}{2} - t)}^{2} t^{2}} = \frac{1}{2} {(\frac{1 - \frac{t^{2}}{2}}{\frac{π}{2} - t})}^{2}

\Rightarrow \lim_{t \to 0} h (t) = \frac{1}{2} (\frac{4}{π^{2}}) = \frac{2}{π^{2}} = \lim_{x \to \frac{π}{2}} g (x)