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lim-x-1-2-x-x-




Question Number 175765 by sciencestudent last updated on 06/Sep/22
lim_(x→∞) (1+(2/x))^(−x) =?
$${li}\underset{{x}\rightarrow\infty} {{m}}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{−{x}} =? \\ $$
Answered by Ar Brandon last updated on 06/Sep/22
lim_(x→∞) (1+(2/x))^(−x)   =lim_(x→∞) ((1+(2/x))^(x/2) )^(−2) =e^(−2) =(1/e^2 )
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{−{x}} \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\left(\mathrm{1}+\frac{\mathrm{2}}{{x}}\right)^{\frac{{x}}{\mathrm{2}}} \right)^{−\mathrm{2}} ={e}^{−\mathrm{2}} =\frac{\mathrm{1}}{{e}^{\mathrm{2}} } \\ $$

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