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Differential Calculus

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 Joined: 1 Apr 2012 Post: 28
25 May 2012 14:26:59 IST
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Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Differential Calculus

Limx->0(tanx/ x)^( 1/x^2) ans-e^1/3. How. Explain? ?????

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Joined: 19 Jan 2008
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27 May 2012 12:20:49 IST
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$\hspace{-16}\bf{\lim_{x\rightarrow 0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^2}}}\\\\\\ \bf{\ast \tan (x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+................}\\\\\\ \bf{\lim_{x\rightarrow 0}\frac{\tan (x)}{x}=\lim_{x\rightarrow 0}\;\; 1+\frac{x^2}{3}+\frac{2x^4}{15}+........}\\\\\\ So \bf{\lim_{x\rightarrow 0}\frac{\tan(x)}{x}\approx \left(1+\frac{x^2}{3}\right)}\\\\\\ So \bf{\lim_{x\rightarrow 0}\left(1+\frac{x^2}{3}\right)^{\frac{1}{x^2}}=e^{\frac{1}{3}}}\\\\\\ Using \bf{\lim_{x\rightarrow 0}\left(1+\frac{x}{\lambda}\right)^{\frac{1}{x}}=e^{\lambda}}$

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