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Math Help - Limits Help!

  1. #1
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    Limits Help!

    Use the definition of limits, find the delta which corresponds to the epsilon givn the limit:
    lim ( 1+x ) ^ 1/x = e ; given the epsilon = .001
    x--->0

    my solns:
    |(1+x)^1/x - e| < .001
    so,
    -.001 < (1+x)^1/x - e < .001
    and,
    e - .001 < (1+x)^(1/x) < e + .001
    .999 < (1+x)^(1/x) < 1.0010005 <=== This is wer i got stuck on how to.. ??



    /epsilon
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  2. #2
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    Quote Originally Posted by ^_^Engineer_Adam^_^ View Post
    Use the definition of limits, find the delta which corresponds to the epsilon givn the limit:
    lim ( 1+x ) ^ 1/x = e ; given the epsilon = .001
    x--->0

    my solns:
    |(1+x)^1/x - e| < .001
    so,
    -.001 < (1+x)^1/x - e < .001
    and,
    e - .001 < (1+x)^(1/x) < e + .001
    .999 < (1+x)^(1/x) < 1.0010005 <=== This is wer i got stuck on how to.. ??



    /epsilon
    We need to find a \delta>0 such that, for all
    |x|<\delta we have that,
    \left|\left(1+x\right)^{1/x}-e\right|<.001
    The delta inequality is more conviently written as,
    -\delta <x<\delta
    The second inequality can be written as,
    -.001<\left(1+x\right)^{1/x}-e<.001
    Thus,
    e-.001<\left(1+x\right)^{1/x}<e+.001
    Here is the problem, I have no idea how to do this delta-epsilon. In fact, this limit is completely proved otherwise.
    Below I have attach a beautiful hand drawn diagram that shows for \delta=.0001 the delta-epsilon statement is true.

    The thin red line shows the curve y=(1+x)^{1/x}
    The thick blue and purple region show that it was squeezed between that area.
    Attached Thumbnails Attached Thumbnails Limits Help!-picture9.gif  
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