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Question Number 199597 by mnjuly1970 last updated on 05/Nov/23

$$$$$$\mathrm{calculate}...$$$$$$$$\mathrm{Q}:{\lim }_{x\to \frac{\pi }{4}}{(1+sin\left(x\right)-cos\left(x\right))}^{tan\left(2x\right)}=?$$$$\bullet \mathcal{N}ice-\mathcal{M}athematics\bullet$$$$$$

Answered by witcher3 last updated on 05/Nov/23

$${\mathrm{e}}^{\mathrm{tg}\left(2\mathrm{x}\right)\ln (1+\sin \left(\mathrm{x}\right)-\cos \left(\mathrm{x}\right))}$$$$\ln (1+\sin \left(\mathrm{x}\right)-\cos \left(\mathrm{x}\right))$$$$\mathrm{x}=\frac{\pi }{4}+\mathrm{y}$$$$\mathrm{tg}\left(2\mathrm{x}\right)=-\cot \left(2\mathrm{y}\right)$$$$1+\sin \left(\mathrm{x}\right)-\cos \left(\mathrm{x}\right)=1+\frac{\sin \left(\mathrm{y}\right)+\cos \left(\mathrm{y}\right)}{\sqrt{2}}-\frac{\cos \left(\mathrm{y}\right)-\sin \left(\mathrm{y}\right)}{\sqrt{2}}$$$$.1+\sqrt{2}\sin \left(\mathrm{y}\right)=1+\mathrm{y}\sqrt{2}+\mathrm{o}\left(\mathrm{y}\right)$$$$\ln (1+\sin \left(\mathrm{y}\right)\sqrt{2})=\mathrm{y}\sqrt{2}+\mathrm{o}(\mathrm{y}\{$$$$-\cot \left(2\mathrm{y}\right)\ln (1+\mathrm{y}\sqrt{2)}\{\sim -\frac{1}{\sqrt{2}}$$$$={\mathrm{e}}^{-\frac{1}{\sqrt{2}}}$$$$$$

Commented by mnjuly1970 last updated on 05/Nov/23

Commented by witcher3 last updated on 06/Nov/23

$$\mathrm{thank}\mathrm{You}\mathrm{Withe}\mathrm{pleasur}$$

Answered by MM42 last updated on 05/Nov/23

$${lim}_{x\to \frac{\pi }{4}}{e}^{(sinx-cosx)\times tan2x}$$$$={lim}_{x\to \frac{\pi }{4}}{e}^{\frac{-cos2x}{(sinx+cosx)}\times \frac{sin2x}{cos2x}}$$$$=e^{-\frac{\sqrt{2}}{2}}✓$$$$$$

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