Question Number 71894 by 20190927 last updated on 21/Oct/19
$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{tanx}} \\ $$
Commented by kaivan.ahmadi last updated on 21/Oct/19
$$={e}^{{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \left({sinx}−\mathrm{1}\right){tanx}} ={e}^{{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \frac{{sinx}−\mathrm{1}}{{cotx}}} =\:\: \\ $$$${e}^{{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{2}}} \frac{{cosx}}{−\left(\mathrm{1}+{cot}^{\mathrm{2}} {x}\right)}} ={e}^{\mathrm{0}} =\mathrm{1} \\ $$