lim-x-pi-2-tan-2-x-x-pi-2-cos-2-x-

Question Number 105661 by bobhans last updated on 30/Jul/20

\lim_{x \to π / 2} \tan^{2} x \sqrt{(x - \frac{π}{2}) - \cos^{2} x} ?

Commented by Dwaipayan Shikari last updated on 30/Jul/20

l i m i t {d o e s n}^{'} t e x i s t

Answered by bemath last updated on 30/Jul/20

s e t x = \frac{π}{2} + m

\lim_{m \to 0} \frac{\sqrt{m - \sin^{2} (\frac{π}{2} + m)}}{\cot^{2} (\frac{π}{2} + m)}

\lim_{m \to 0} \frac{\sqrt{m - \cos^{2} (m)}}{\tan^{2} m} = \sqrt{m} \lim_{m \to 0} \frac{\sqrt{1 - \frac{\cos^{2} m}{m}}}{\tan^{2} m}

= \sqrt{m} \lim_{x \to 0} \frac{1 - \frac{\cos^{2} m}{2 m}}{\tan^{2} m} = \sqrt{m} \lim_{m \to 0} \frac{1 - {(1 - \frac{m^{2}}{2})}^{2}}{}

Answered by Dwaipayan Shikari last updated on 30/Jul/20

\lim_{x \to \frac{π^{-}}{2}} {t a n}^{2} x \sqrt{x - \frac{π}{2} - h - {c o s}^{2} x} \to i \infty {(h \to 0) (L . H . S)

(R . H . S) \lim_{x \to \frac{π^{+}}{2}} {t a n}^{2} x \sqrt{x - \frac{π}{2} + h - {c o s}^{2} x} \to \infty

R . H . S \neq L . H . S