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Question Number 175362 by cortano1 last updated on 28/Aug/22

$$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\sin x-\cos x}{\tan (\frac{\pi }{8}-\frac{x}{2})}=?$$

Commented by infinityaction last updated on 28/Aug/22

$$\underset{x\to \frac{\pi }{4}}{\lim }\frac{-\sqrt{2}\sin (\pi /4-x)}{\sin (\frac{\pi }{8}-\frac{x}{2})\sec (\frac{\pi }{8}-\frac{x}{2})}$$$$\underset{x\to \frac{\pi }{4}}{\lim }\frac{-2\sqrt{2}\cancel{\sin (\frac{\pi }{8}-\frac{x}{2})}\cos (\frac{\pi }{8}-\frac{x}{2})}{\cancel{\sin (\frac{\pi }{8}-\frac{x}{2})}}$$$$-2\sqrt{2}$$

Commented by CElcedricjunior last updated on 28/Aug/22

$$\underset{x\to \frac{\mathit{\pi }}{4}}{\lim }\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\mathit{t}\mathit{a}\mathit{n}(\frac{\mathit{\pi }}{8}-\frac{\mathit{x}}{2})}=\frac{0}{0}=F\mathit{I}$$$$\mathit{e}\mathit{n}\mathit{d}\mathit{a}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{h}\mathit{o}\mathit{s}\mathit{p}\mathit{i}\mathit{t}\mathit{a}\mathit{l}$$$$\underset{x\to \frac{\mathit{\pi }}{4}}{\lim }\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{x}+\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{-\frac{1}{2}[1+\mathit{t}\mathit{a}\stackrel{2}{\mathit{n}}(\frac{\mathit{\pi }}{8}-\frac{\mathit{x}}{2})]}=\frac{\sqrt{2}}{-\frac{1}{2}}$$$$\underset{x\to \frac{\mathit{\pi }}{4}}{\lim }\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\mathit{c}\mathit{o}\mathit{s}\mathit{x}}{\mathit{t}\mathit{a}\mathit{n}(\frac{\mathit{\pi }}{8}-\frac{\mathit{x}}{2})}=-2\sqrt{2}$$$$$$$$.........\mathit{l}\mathit{e}c\acute{el}\stackrel{`}{ebre}cedricjunior.........$$

Answered by BaliramKumar last updated on 28/Aug/22

$$\underset{x\to \frac{\pi }{4}}{\lim }\frac{\frac{d}{dx}(sinx-cosx)}{\frac{d}{dx}tan(\frac{\pi }{8}-\frac{x}{2})}=\underset{x\to \frac{\pi }{4}}{\lim }\frac{cosx+sinx}{-\frac{1}{2}{sec}^2(\frac{\pi }{8}-\frac{x}{2})}$$$$\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{\frac{-1}{2}\times 1}=-2\sqrt{2}$$

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