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Evaluate: $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$$

attemp: Take $P=\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$ . Then taking log both side .$$\ln P=-\lim_{n \to \infty}n\ln [(1+\frac{1}{n})^n-(1+\frac{1}{n})]$$.Then stuck. Please help.

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Do you know what $\displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$ is? –  Michael Albanese Dec 21 '12 at 12:33
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yes.It is equal to e. –  A.D Dec 21 '12 at 12:34
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2 Answers 2

$(1+\frac{1}{n})^n$ tends to $e$ as $n$ increases, so for big enough $n$ it's between $2$ and $3$. Then you can continue with squeeze theorem for example. We have $$\frac{3}{2} \le 2 - (1+\frac{1}{n})\le(1+\frac{1}{n})^n-(1+\frac{1}{n}) \le 3 -(1+\frac{1}{n})\le 2$$ for big $n$. We see that $\lim_{n \to \infty}(\frac{3}{2})^{-n} = \lim_{n \to \infty}(2)^{-n} = 0$, so $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$ is also $0$

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Please explain it more. –  A.D Dec 21 '12 at 12:41
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What you have done is wrong because you assumed the limit existed.

We must evaluate $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}= \lim_{n \to \infty}e^{\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)} $$ Remember $$\lim_{n \to \infty}(1+\frac{1}{n})^n=e$$ Thus, because $e>2$ $$\lim_{n \to \infty}\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)=\log(e-1)(-\infty)=-\infty $$ By continuity of $e^x$, $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}= \lim_{n \to \infty}e^{\log[(1+\frac{1}{n})^n-(1+\frac{1}{n})](-n)}=e^{-\infty}=0 $$

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@Jabali: You wanted Adam to explain more but you have a complete answer here! :-) –  B. S. Dec 21 '12 at 12:50
    
I don't understand adam's word.So I asked to explain.But I don't understand here how I conclude that limit does not exist. –  A.D Dec 21 '12 at 13:02
    
@DonAntonio Oh my! That's a serious mistake. I will correct this immediately –  Nameless Dec 21 '12 at 14:53
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