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

Question Number 114637 by bobhans last updated on 20/Sep/20

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

Answered by bemath last updated on 20/Sep/20

s e t t i n g x = \frac{π}{2} + p \to 2 x = π + 2 p

\lim_{p \to 0} \frac{\frac{π}{2} - \cos^{- 1} (2 p)}{1 - \sin^{- 1} (\frac{π + 2 p}{π})} =

\lim_{p \to 0} \frac{[\frac{2}{\sqrt{1 - 4 p^{2}}}]}{- [\frac{\frac{2}{π}}{\sqrt{1 - {(\frac{π + 2 p}{π})}^{2}}}]} =

\lim_{p \to 0} \frac{2}{\sqrt{1 - 4 p^{2}}} \times (- \frac{2}{π \sqrt{1 - {(\frac{π + 2 p}{π})}^{2}}}) = \infty

Commented by bobhans last updated on 20/Sep/20

g a v e k u d o s

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