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Question Number 280 by arnav last updated on 25/Jan/15

$$\mathrm{Evaluate}\underset{x\to \pi /4}{\lim }\frac{\cos x-\sin x}{(\pi /4-x)(\cos x+\sin x)}$$

Answered by 123456 last updated on 18/Dec/14

$$\underset{x\to \pi /4}{\lim }\frac{\cos x-\sin x}{(\frac{\pi }{4}-x)(\cos x+\sin x)}\to \frac{0}{0}$$$$=\underset{x\to \pi /4}{\lim }\frac{-\sin x-\cos x}{-(\cos x+\sin x)+(\frac{\pi }{4}-x)(-\sin x+\cos x)}$$$$=\frac{-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}}{-(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2})+(\frac{\pi }{4}-\frac{\pi }{4})(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2})}$$$$=1$$

Answered by 123456 last updated on 22/Dec/14

$$=\underset{x\to \pi /4}{\lim }\frac{\cos x-\sin x}{(\frac{\pi }{4}-x)(\cos x+\sin x)}$$$$=\underset{x\to \pi /4}{\lim }\frac{\sin (\frac{\pi }{4}-x)}{(\frac{\pi }{4}-x)\sin (\frac{\pi }{4}+x)}\cdot \frac{\sqrt{2}}{\sqrt{2}}$$$$=\underset{x\to \pi /4}{\lim }\frac{\sin (\frac{\pi }{4}-x)}{\frac{\pi }{4}-x}\underset{x\to \pi /4}{\lim }\frac{1}{\sin (\frac{\pi }{4}+x)}$$$$=\underset{y\to 0}{\lim }\frac{\sin y}{y}\cdot \frac{1}{\sin (\frac{\pi }{4}+\frac{\pi }{4})}$$$$=1\cdot \frac{1}{\sin \frac{\pi }{2}}=1$$

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