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Question Number 31524 by abdo imad last updated on 09/Mar/18

find lim_(n→∞) (1+sin((1/n)))^n .

$${find}\:{lim}_{{n}\rightarrow\infty} \left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{{n}} . \\ $$

Commented by abdo imad last updated on 12/Mar/18

let put A_n  =(1+sin((1/n)))^n  we have   ln(A_n )=nln(1+sin((1/n)))  but sin((1/n))=(1/n) +o((1/n))and  ln(1+sin((1/n))) = (1/n) +o((1/n)) ⇒ nln(1+sin((1/n)))=1+o(1)  so lim_(n→∞) ln(A_n )= 1 ⇒ lim_(n→∞) A_n = e .

$${let}\:{put}\:{A}_{{n}} \:=\left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{{n}} \:{we}\:{have}\: \\ $$$${ln}\left({A}_{{n}} \right)={nln}\left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)\:\:{but}\:{sin}\left(\frac{\mathrm{1}}{{n}}\right)=\frac{\mathrm{1}}{{n}}\:+{o}\left(\frac{\mathrm{1}}{{n}}\right){and} \\ $$$${ln}\left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)\:=\:\frac{\mathrm{1}}{{n}}\:+{o}\left(\frac{\mathrm{1}}{{n}}\right)\:\Rightarrow\:{nln}\left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)=\mathrm{1}+{o}\left(\mathrm{1}\right) \\ $$$${so}\:{lim}_{{n}\rightarrow\infty} {ln}\left({A}_{{n}} \right)=\:\mathrm{1}\:\Rightarrow\:{lim}_{{n}\rightarrow\infty} {A}_{{n}} =\:{e}\:. \\ $$

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