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Question Number 194852 by mustafazaheen last updated on 17/Jul/23

$$$$$$\underset{x\to 0}{\lim }{\left(\mathrm{cosx}\right)}^{\log \left(\mathrm{x}\right)}=?$$$$$$

Answered by MM42 last updated on 17/Jul/23

$${lim}_{x\to 0}{\left(cosx\right)}^{logx}=e^{{lim}_{x\to 0}(cosx-1)logx}$$$$=e^{{lim}_{x\to 0}\frac{\frac{lnx}{ln10}}{\frac{-1}{x^2}}}=e^{{lim}_{x\to 0}\frac{\frac{1}{ln10}\times \frac{1}{x}}{\frac{2}{x^3}}}$$$$=e^{{lim}_{x\to 0}\frac{x^2}{2ln10}}=1$$$$$$

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