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

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

    I am having trouble showing that \lim_{x\to \infty}(1+\frac{1}{x})^x = e.
    All I have gotten is that I somehow have to use the sequence e_{n}=(1+\frac{1}{n})^n. Im not really sure where to start though.
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  2. #2
    MHF Contributor chisigma's Avatar
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    One possible way is to valuate...

    \ln (1+\frac{1}{x})^{x} = x\cdot \ln (1+\frac{1}{x})= x \cdot (\frac{1}{x} - \frac{1}{2x^{2}} + \frac{1}{3x^{3}} - ...) (1)

    From (1) is evident that...

    \lim_{x \rightarrow \infty} x\cdot \ln (1+\frac{1}{x}) =1 (2)

    Kind regards

    \chi \sigma
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  3. #3
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    how could I prove it without using l'hopitals rule, i.e. using sequences
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