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Question Number 132138 by benjo_mathlover last updated on 11/Feb/21

lim_(n→∞) ((((1+(1/n))^n )/((1−(1/n))^n )) − e^2 )n^2 =?

$$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }\:−\:\mathrm{e}^{\mathrm{2}} \:\right)\mathrm{n}^{\mathrm{2}} =? \\ $$

Answered by EDWIN88 last updated on 11/Feb/21

L=e^2 ×lim_(x→0) ((ln (1+x)−ln (1−x)−2x)/x^3 ) L=e^2 ×lim_(x→0) (((1/(1+x))+(1/(1−x))−2)/(3x^2 )) L=e^2 ×lim_(x→0) ((2−2(1−x^2 ))/((1+x)(1−x)3x^2 )) L=e^2 ×lim_(x→0) ((2x^2 )/(3x^2 )) = ((2e^2 )/3)

$$\mathrm{L}=\mathrm{e}^{\mathrm{2}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{x}\right)−\mathrm{ln}\:\left(\mathrm{1}−\mathrm{x}\right)−\mathrm{2x}}{\mathrm{x}^{\mathrm{3}} } \\ $$$$\mathrm{L}=\mathrm{e}^{\mathrm{2}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}}−\mathrm{2}}{\mathrm{3x}^{\mathrm{2}} } \\ $$$$\mathrm{L}=\mathrm{e}^{\mathrm{2}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}−\mathrm{2}\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)}{\left(\mathrm{1}+\mathrm{x}\right)\left(\mathrm{1}−\mathrm{x}\right)\mathrm{3x}^{\mathrm{2}} } \\ $$$$\mathrm{L}=\mathrm{e}^{\mathrm{2}} ×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2x}^{\mathrm{2}} }{\mathrm{3x}^{\mathrm{2}} }\:=\:\frac{\mathrm{2e}^{\mathrm{2}} }{\mathrm{3}} \\ $$

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