Question Number 199597 by mnjuly1970 last updated on 05/Nov/23
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{calculate}… \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{Q}\::\:\:\:\:\:\:\mathrm{lim}_{\:{x}\rightarrow\:\frac{\pi}{\mathrm{4}}} \:\:\left(\:\:\mathrm{1}\:+\:\:{sin}\left({x}\right)\:−{cos}\left({x}\right)\:\right)^{\:{tan}\left(\mathrm{2}{x}\right)} \:=\:\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bullet\:\mathscr{N}{ice}\:−\:\mathscr{M}{athematics}\:\bullet\: \\ $$$$\:\:\:\:\:\:\: \\ $$
Answered by witcher3 last updated on 05/Nov/23
$$\mathrm{e}^{\mathrm{tg}\left(\mathrm{2x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)−\mathrm{cos}\left(\mathrm{x}\right)\right)} \\ $$$$\mathrm{ln}\left(\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)−\mathrm{cos}\left(\mathrm{x}\right)\right) \\ $$$$\mathrm{x}=\frac{\pi}{\mathrm{4}}+\mathrm{y} \\ $$$$\mathrm{tg}\left(\mathrm{2x}\right)=−\mathrm{cot}\left(\mathrm{2y}\right) \\ $$$$\mathrm{1}+\mathrm{sin}\left(\mathrm{x}\right)−\mathrm{cos}\left(\mathrm{x}\right)=\mathrm{1}+\frac{\mathrm{sin}\left(\mathrm{y}\right)+\mathrm{cos}\left(\mathrm{y}\right)}{\:\sqrt{\mathrm{2}}}−\frac{\mathrm{cos}\left(\mathrm{y}\right)−\mathrm{sin}\left(\mathrm{y}\right)}{\:\sqrt{\mathrm{2}}} \\ $$$$.\mathrm{1}+\sqrt{\mathrm{2}}\mathrm{sin}\left(\mathrm{y}\right)=\mathrm{1}+\mathrm{y}\sqrt{\mathrm{2}}+\mathrm{o}\left(\mathrm{y}\right) \\ $$$$\mathrm{ln}\left(\mathrm{1}+\mathrm{sin}\left(\mathrm{y}\right)\sqrt{\mathrm{2}}\right)=\mathrm{y}\sqrt{\mathrm{2}}+\mathrm{o}\left(\mathrm{y}\left\{\right.\right. \\ $$$$−\mathrm{cot}\left(\mathrm{2y}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{y}\sqrt{\left.\mathrm{2}\right)}\left\{\sim−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right.\right. \\ $$$$=\mathrm{e}^{−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}} \\ $$$$ \\ $$
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
$${lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:{e}^{\left({sinx}−{cosx}\right)×{tan}\mathrm{2}{x}} \\ $$$$={lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:{e}^{\frac{−{cos}\mathrm{2}{x}}{\left({sinx}+{cosx}\right)}×\frac{{sin}\mathrm{2}{x}}{{cos}\mathrm{2}{x}}} \\ $$$$={e}^{−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}} \:\:\checkmark \\ $$$$ \\ $$