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Question-144513




Question Number 144513 by mnjuly1970 last updated on 26/Jun/21
Answered by Olaf_Thorendsen last updated on 26/Jun/21
(1+(1/x))^(−x)  = e^(−xln(1+(1/x)))  ∼_∞  e^(−x((1/x)−(1/(2x^2 ))))   (1+(1/x))^(−x)  ∼_∞  (1/e)e^(1/(2x))     (1−(1/x))^x  = e^(xln(1−(1/x)))  ∼_∞  e^(x(−(1/x)−(1/(2x^2 ))))   (1−(1/x))^x  ∼_∞  (1/e)e^(−(1/(2x)))     f(x) = x[(1+(1/x))^(−x) −(1−(1/x))^x ]  f(x) ∼_∞  (x/e)[e^(1/(2x)) −e^(−(1/(2x))) ]  f(x) ∼_∞  ((2x.sinh((1/(2x))))/e) ∼_∞  ((2x.(1/(2x)))/e) = (1/e)
$$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{−{x}} \:=\:{e}^{−{x}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)} \:\underset{\infty} {\sim}\:{e}^{−{x}\left(\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)} \\ $$$$\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{−{x}} \:\underset{\infty} {\sim}\:\frac{\mathrm{1}}{{e}}{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}} \\ $$$$ \\ $$$$\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)^{{x}} \:=\:{e}^{{x}\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)} \:\underset{\infty} {\sim}\:{e}^{{x}\left(−\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)} \\ $$$$\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)^{{x}} \:\underset{\infty} {\sim}\:\frac{\mathrm{1}}{{e}}{e}^{−\frac{\mathrm{1}}{\mathrm{2}{x}}} \\ $$$$ \\ $$$${f}\left({x}\right)\:=\:{x}\left[\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{−{x}} −\left(\mathrm{1}−\frac{\mathrm{1}}{{x}}\right)^{{x}} \right] \\ $$$${f}\left({x}\right)\:\underset{\infty} {\sim}\:\frac{{x}}{{e}}\left[{e}^{\frac{\mathrm{1}}{\mathrm{2}{x}}} −{e}^{−\frac{\mathrm{1}}{\mathrm{2}{x}}} \right] \\ $$$${f}\left({x}\right)\:\underset{\infty} {\sim}\:\frac{\mathrm{2}{x}.\mathrm{sinh}\left(\frac{\mathrm{1}}{\mathrm{2}{x}}\right)}{{e}}\:\underset{\infty} {\sim}\:\frac{\mathrm{2}{x}.\frac{\mathrm{1}}{\mathrm{2}{x}}}{{e}}\:=\:\frac{\mathrm{1}}{{e}} \\ $$

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