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Question Number 101981 by Rohit@Thakur last updated on 05/Jul/20

$$li\underset{x\to 0}{m}{\left(\frac{tanx}{x}\right)}^{\frac{1}{x^2}}$$

Answered by Dwaipayan Shikari last updated on 06/Jul/20

$$\underset{x\to 0}{\lim }{\left(\frac{tanx}{x}\right)}^{\frac{1}{x^2}}=y$$$$\underset{x\to 0}{\lim }\frac{1}{x^2}log\left(\frac{tanx}{x}\right)=logy$$$$\underset{x\to 0}{\lim }\frac{1}{x^2}log(1+\frac{tanx-x}{x})=logy$$$$\underset{x\to 0}{\lim }\frac{1}{x^2}log(1+\frac{\frac{x^3}{3}+x-x}{x})=logy$$$$\underset{x\to 0}{\lim }\frac{1}{3}\frac{log(1+\frac{x^2}{3})}{\frac{x^2}{3}}=logy$$$$logy=\frac{1}{3}\Rightarrow y=e^{\frac{1}{3}}$$$$$$

Answered by john santu last updated on 06/Jul/20

$$\underset{x\to 0}{\lim }{(1+(\frac{\tan x}{x}-1))}^{\frac{1}{x^2}}=$$$$e^{\underset{x\to 0}{\lim }\left(\frac{\tan x-x}{x}\right).\frac{1}{x^2}}=e^{\underset{x\to 0}{\lim }\left(\frac{\tan x-x}{x^3}\right)}$$$$=e^{\underset{x\to 0}{\lim }\left(\frac{(x+\frac{1}{3}{x}^3)-x}{x^3}\right)}=e^{\frac{1}{3}}=\sqrt[3]{e}$$$$(JS⊛)$$

Answered by mathmax by abdo last updated on 06/Jul/20

$$\mathrm{let}\mathrm{f}\left(\mathrm{x}\right)={\left(\frac{\mathrm{tanx}}{\mathrm{x}}\right)}^{\frac{1}{{\mathrm{x}}^2}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{1}{{\mathrm{x}}^2}\ln \left(\frac{\mathrm{tanx}}{\mathrm{x}}\right)}$$$$\mathrm{we}\mathrm{have}\mathrm{tanx}=\tan 0+\frac{\mathrm{x}}{1!}{\tan }^{{}^{\prime }}\left(0\right)+\frac{{\mathrm{x}}^2}{2!}{\tan }^{\left(2\right)}\left(0\right)+\frac{{\mathrm{x}}^3}{3!}{\tan }^3\left(0\right)+{\mathrm{x}}^5\}\delta \left(\mathrm{x}\right)$$$$\tan 0=0$$$${\tan }^{{}^{\prime }}\mathrm{x}=1+{\tan }^2\mathrm{x}\Rightarrow {\tan }^{{}^{\prime }}\left(0\right)=1$$$${\tan }^{\left(2\right)}\left(0\right)=2\mathrm{tanx}(1+{\tan }^2\mathrm{x})\Rightarrow {\tan }^{\left(2\right)}\left(0\right)=0$$$${\tan }^{\left(3\right)}\left(0\right)=2(1+{\tan }^2\mathrm{x})(1+{\tan }^2\mathrm{x})+2\mathrm{tanx}\left(2\mathrm{tanx}\right)(1+{\tan }^2\mathrm{x})\Rightarrow$$$${\tan }^{\left(3\right)}\left(0\right)=2\Rightarrow \mathrm{tanx}=\mathrm{x}+\frac{{\mathrm{x}}^3}{3}+\mathrm{o}\left({\mathrm{x}}^5\right)\Rightarrow \frac{\mathrm{tanx}}{\mathrm{x}}=1+\frac{{\mathrm{x}}^2}{3}+\mathrm{o}\left({\mathrm{x}}^4\right)\Rightarrow$$$$\ln \left(\frac{\mathrm{tanx}}{\mathrm{x}}\right)=\ln (1+\frac{{\mathrm{x}}^2}{3}+\mathrm{o}\left({\mathrm{x}}^4\right)))\sim \frac{{\mathrm{x}}^2}{3}\Rightarrow \frac{1}{{\mathrm{x}}^2}\ln \left(\frac{\mathrm{tanx}}{\mathrm{x}}\right)\sim \frac{1}{3}\Rightarrow$$$${\lim }_{\mathrm{x}\to 0}\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\frac{1}{3}}{=}^3\sqrt{\mathrm{e}}$$

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