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(1+x)^(1/z) - Limit as x approaches 0

  1. May 1, 2012 #1
    I was given this as a practice question, and assumed that the answer would be 0.

    The answer is e, but no explanation is given and I cannot figure out why this is. z is undefined in the question.

    If anyone could shed some light it would be greatly appreciated.
     
  2. jcsd
  3. May 1, 2012 #2

    Curious3141

    User Avatar
    Homework Helper

    There shouldn't be a z, it should simply be replaced by x.

    [tex]\lim_{x \rightarrow 0} {(1 + x)}^{\frac{1}{x}} = e[/tex]

    That's actually one of the definitions for e (the base of natural logarithms). If you want a fairly elementary (but not very rigorous) way of proving it, consider putting [itex]y = \frac{1}{x}[/itex], from which you get the equivalent definition:

    [tex]\lim_{y \rightarrow \infty} {(1 + \frac{1}{y})}^y = e[/tex]

    and apply Binomial theorem to the LHS. Expand and consider the limit as y tends to infinity. Now compare that with the Taylor series for e (e1).
     
  4. May 1, 2012 #3
    Thankyou very much!
     
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