prove-that-lim-x-pi-2-tan-x-2-1-x-pi-2-1-

Question Number 195325 by mathlove last updated on 30/Jul/23

p r o v e t h a t

\lim_{x \to \frac{π}{2}} \frac{t a n (\frac{x}{2}) - 1}{x - \frac{π}{2}} = 1

Answered by BaliramKumar last updated on 30/Jul/23

\lim_{x \to \frac{π}{2}} \frac{\frac{d}{dx} (t a n (\frac{x}{2}) - 1)}{\frac{d}{dx} (x - \frac{π}{2})} = \frac{\sec^{2} (\frac{x}{2}) \cdot \frac{1}{2}}{1}

\frac{1}{2} \sec^{2} (\frac{π}{4}) = \frac{1}{2} {(\sqrt{2})}^{2} = 1

Answered by som(math1967) last updated on 30/Jul/23

\underset{x \to \frac{π}{2}}{l i m} \frac{s i n \frac{x}{2} - c o s \frac{x}{2}}{c o s \frac{x}{2} (x - \frac{π}{2})}

\underset{x \to \frac{π}{2}}{l i m} \frac{\sqrt{2} (\frac{1}{\sqrt{2}} s i n \frac{x}{2} - \frac{1}{\sqrt{2}} c o s \frac{x}{2})}{2 c o s \frac{x}{2} (\frac{x}{2} - \frac{π}{4})}

\underset{x \to \frac{π}{2}}{l i m} \frac{\sqrt{2} s i n (\frac{x}{2} - \frac{π}{4})}{2 \times \frac{1}{\sqrt{2}} (\frac{x}{2} - \frac{π}{4})}

\underset{x \to \frac{π}{2}}{l i m} \frac{s i n (\frac{x}{2} - \frac{π}{4})}{(\frac{x}{2} - \frac{π}{4})}

l e t (\frac{x}{2} - \frac{π}{4}) = t

x \to \frac{π}{2} \Rightarrow \frac{x}{2} \to \frac{π}{4} ∴ (\frac{x}{2} - \frac{π}{4}) \to 0

∴ \underset{t \to 0}{l i m} \frac{s i n t}{t} = 1

Commented by mathlove last updated on 30/Jul/23

t n k s