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Question Number 127213 by Study last updated on 27/Dec/20

$$li\underset{x\to 0}{m}{\left(cosx\right)}^{logx}=???bysandiwichrule$$

Commented by Study last updated on 27/Dec/20

$$helpme$$

Commented by Study last updated on 28/Dec/20

$$pleaseineedyourhelp$$

Answered by mathmax by abdo last updated on 28/Dec/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)={\left(\mathrm{cosx}\right)}^{\mathrm{logx}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{logxlog}\left(\mathrm{cosx}\right)}$$$$\log \left(\mathrm{cosx}\right)\sim \log (1-\frac{{\mathrm{x}}^2}{2})\sim -\frac{{\mathrm{x}}^2}{2}\Rightarrow \mathrm{logx}.\log \left(\mathrm{cosx}\right)\sim -\frac{{\mathrm{x}}^2}{2}\mathrm{logx}\to 0(\mathrm{x}\to {\mathrm{o}}^{+})\Rightarrow$$$${\lim }_{\mathrm{x}\to 0^{+}}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^0=1$$

Commented by Study last updated on 28/Dec/20

$$whocansolvebysandwichrule???$$

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