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Math Help - limit of (1+1/x)^x

  1. #1
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    limit of (1+1/x)^x

    how do we show the limit of (1+1/x)^x is e (as x tends to infinity) by bounding it above and showing that it is an increasing sequence?
    Thanks for any help
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  2. #2
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    See natural log base and the links on that page on PlanetMath. They consider a sequence (1+1/n)^n, not a function, however.
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  3. #3
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by hmmmm View Post
    how do we show the limit of (1+1/x)^x is e (as x tends to infinity) by bounding it above and showing that it is an increasing sequence?
    Thanks for any help
    Let's consider the function \lambda(x) = \ln \{(1 + \frac{1}{x})^{x}\}= x\ \ln (1+\frac{1}{x}). For |x|>1 is...

    \displaystyle \lambda (x) = 1 - \frac{1}{2\ x} + \frac{1}{3\ x^{2}} - ... (1)

    ... so that is...

    \displaystyle \lim_{x \rightarrow \infty} \lambda(x) = 1 \implies \lim_{x \rightarrow \infty} (1+\frac{1}{x})^{x}= e (2)

    From (1) it is easy to derive that if x_{2}>x_{1}>1 is \lambda(x_{2}) > \lambda(x_{1}) so that \lambda (x) is an increasing function...

    Kind regards

    \chi \sigma
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