Question Number 194852 by mustafazaheen last updated on 17/Jul/23
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{cosx}\right)^{\mathrm{log}\left(\mathrm{x}\right)} =? \\ $$$$ \\ $$
Answered by MM42 last updated on 17/Jul/23
$${lim}_{{x}\rightarrow\mathrm{0}} \:\left({cosx}\right)^{{logx}} ={e}^{{lim}_{{x}\rightarrow\mathrm{0}} \:\left({cosx}−\mathrm{1}\right){logx}} \\ $$$$={e}^{{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{\frac{{lnx}}{{ln}\mathrm{10}}}{\frac{−\mathrm{1}}{{x}^{\mathrm{2}} }}} \:={e}^{{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\frac{\mathrm{1}}{{ln}\mathrm{10}}×\frac{\mathrm{1}}{{x}}}{\frac{\mathrm{2}}{{x}^{\mathrm{3}} }}} \\ $$$$={e}^{{lim}_{{x}\rightarrow\mathrm{0}} \:\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}{ln}\mathrm{10}}} =\mathrm{1} \\ $$$$ \\ $$