prove that lim_(x→(π/2)) ((tan((x/2))−1)/(x−(π/2)))=1


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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

	$t n k s$
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