lim-x-pi-2-x-cot-x-pi-2cos-x-

Question Number 142459 by bramlexs22 last updated on 01/Jun/21

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\lim_{x \to \frac{π}{2}} (\frac{x}{\cot x} - \frac{π}{2 \cos x}) = ?

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Answered by benjo_mathlover last updated on 01/Jun/21

\lim_{x \to \frac{π}{2}} (\frac{x \sin x}{\cos x} - \frac{π}{2 \cos x})

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

= \lim_{x \to \frac{π}{2}} (\frac{2 \sin x + 2 x \cos x}{- 2 \sin x})

= - 1

Answered by mathmax by abdo last updated on 02/Jun/21

f (x) = \frac{x}{cotanx} - \frac{π}{2 cosx} \Rightarrow f (x) = \frac{xsinx}{cosx} - \frac{π}{2 cosx} = \frac{2 xsinx - π}{2 cosx}

changement x = \frac{π}{2} - t give f (x) = f (\frac{π}{2} - t) = \frac{2 (\frac{π}{2} - t) cost - π}{2 sint}

= \frac{(π - 2 t) cost - π}{2 sint} \sim \frac{(π - 2 t) (1 - \frac{t^{2}}{2}) - π}{2 t}

= \frac{π - \frac{π}{2} t^{2} - 2 t + t^{3} - π}{2 t} = - \frac{π}{4} t - 1 + \frac{t^{2}}{2} \to - 1 (t \to 0) \Rightarrow

\lim_{x \to \frac{π}{2}} f (x) = - 1