Question

${lim}_{x\to 0}(\cos x{)}^{\frac{1}{\sin x}}$

Answer 1

${lim}_{x\to 0}(\cos x{)}^{\frac{1}{\sin x}}={lim}_{x\to }(1-(1-cosx){)}^{\frac{1}{sinx}}={lim}_{x\to 0}{(1-2si{n}^2\frac{x}{2})}^{\frac{1}{sinx}}[\because {\sin }^{2}x=\frac{1-\cos 2x}{2}]={lim}_{x\to }(y-2{\sin }^2\frac{x}{2}-1)\times \frac{1}{\sin x}[\because {lim}_{x\to a}\{f\left(x\right){\}}^{g\left(x\right)}=e^{{lim}_{x\to a}f\left\{\right(x)-1\}g\left(x\right)}]=e^{{lim}_{x\to 0}}\left(\frac{-2{\sin }^2\frac{x}{2}}{2sin\frac{x}{2}\cos \frac{x}{2}}\right)=e^{{lim}_{x\to 0}(-\tan \frac{x}{2})}=e^0=1$