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Question Number 42781 by maxmathsup by imad last updated on 02/Sep/18

$$calculate{lim}_{x\to \frac{\pi }{4}}\frac{sin\left(2x\right)sin(x-\frac{\pi }{4})}{sinx-cosx}$$

Commented by maxmathsup by imad last updated on 04/Oct/18

$$letA\left(x\right)=\frac{sin\left(2x\right)sin(x-\frac{\pi }{4})}{sinx-cosx}wehaveA\left(x\right)=\frac{sin\left(2x\right)sin(x-\frac{\pi }{4})}{\sqrt{2}sin(x-\frac{\pi }{4})}$$$$\Rightarrow {lim}_{x\to \frac{\pi }{4}}A\left(x\right)={lim}_{x\to \frac{\pi }{4}}\frac{sin\left(2x\right)}{\sqrt{2}}=\frac{1}{\sqrt{2}}.$$

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Sep/18

$$li\underset{x\to \frac{\mathrm{\Pi }}{4}}{m}\frac{sin2x\times \frac{1}{\sqrt{2}}(sinx-cosx)}{sinx-cosx}$$$$=\frac{1}{\sqrt{2}}\times sin\frac{\mathrm{\Pi }}{2}=\frac{1}{\sqrt{2}}$$$$$$$$$$$$$$$$$$$$$$$$\underset{t\to 0}{\lim }$$$$\underset{t\to 0}{\mathrm{li}}$$

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