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Question Number 107930 by bemath last updated on 13/Aug/20

$$\frac{\mathbb{B}e\mathbb{M}ath}{\bullet \cap \bullet }$$$$\underset{x\to 0}{\lim }{\left(\sin x\right)}^{\frac{1}{\ln \sqrt{x}}}?$$

Answered by bemath last updated on 13/Aug/20

$$\Rightarrow L=\underset{x\to 0}{\lim }{\left(\sin \right)}^{\frac{1}{\ln \sqrt{x}}}=$$$$\ln L=\underset{x\to 0}{\lim }\frac{\ln \left(\sin x\right)}{\ln \sqrt{x}}=2\left[\underset{x\to 0}{\lim }\frac{\ln \left(\sin x\right)}{\ln x}\right]$$$$\ln L=2\left[\underset{x\to 0}{\lim }\frac{\frac{\cos x}{\sin x}}{\frac{1}{x}}\right]=2\underset{x\to 0}{\lim }\left[\frac{x\cos x}{\sin x}\right]$$$$\ln L=2\Rightarrow L=e^2$$$$$$

Answered by Dwaipayan Shikari last updated on 13/Aug/20

$$\frac{1}{log\sqrt{x}}log\left(sinx\right)=logL$$$$\frac{1}{\frac{1}{2}log\left(x\right)}log\left(x\right)=logL(sinx\to x)$$$$2=logL$$$$L=e^2$$

Answered by mathmax by abdo last updated on 13/Aug/20

$$\mathrm{f}\left(\mathrm{x}\right)={\left(\mathrm{sinx}\right)}^{\frac{1}{\ln \left(\sqrt{\mathrm{x}}\right)}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{1}{\ln \left(\sqrt{\mathrm{x}}\right)}\ln \left(\mathrm{sinx}\right)}$$$$={\mathrm{e}}^{\frac{2}{\mathrm{lnx}}\ln \left(\mathrm{sinx}\right)}\sim {\mathrm{e}}^2(\mathrm{x}\sim 0)\Rightarrow {\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^2$$

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