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Homework Help: Epsilon-delta proof of limit definition of e?

  1. Sep 16, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that
    [tex]\lim_{x\rightarrow\ 0} (1+x)^{1/x}=e[/tex]
    by an epsilon-delta proof.


    2. Relevant equations



    3. The attempt at a solution
    I did:
    x < a
    1 + x < 1 + a
    but I couldn't go any further.
     
  2. jcsd
  3. Sep 16, 2010 #2

    jgens

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    It would help to know what definition of e you're using.
     
  4. Sep 16, 2010 #3
    Anything other than the limit one.
     
  5. Sep 16, 2010 #4

    jgens

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    Gold Member

    Well, pick one that you like, and we'll work from there.
     
  6. Sep 16, 2010 #5
    The only other one I really know is:
    [tex]\sum_{n=0}^{\infty} \frac{1}{n!}[/tex]
     
  7. Sep 17, 2010 #6

    jgens

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    Gold Member

    Okay, so we have our definition of e. Now, note that

    [tex]\lim_{x\to 0}(1+x)^{1/x}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}[/tex]

    Notice that you can expand the term on the right using the binomial theorem. If you can get this term to look like the one that you have for e, then it should be simple to show that their difference can be made as small as desired. Use this as the basis for your proof.
     
  8. Sep 17, 2010 #7
    Thank you. I understand it now.
     
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