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Math Help - Interesting Limit

  1. #1
    Senior Member
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    Interesting Limit

    Evaluate: <br /> <br />
\lim_{x\rightarrow 0}\left(\frac{(1 + x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}}<br />
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  2. #2
    Super Member fardeen_gen's Avatar
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    One of the first I had encountered when I started limits.

    Answer:
    Spoiler:
    e^{-\frac{1}{2}}


    I leave it to the other members to see if they know how to do it.
    HINT:
    Spoiler:
    Use a = e^{\ln a}
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  3. #3
    Super Member Random Variable's Avatar
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    My attempt:

    first find  \lim_{x \to 0} \ \ln \Bigg( \Big( \frac {(1+x)^{1 \over x}}{e} \Big)^{1 \over x} \Bigg)

     = \lim_{x \to 0} \ \frac {1}{x} \Big( \ln (1+x)^{1 \over x} - \ln(e) \Big)

     = \lim_{x \to 0} \ \frac {1}{x} \Big( \frac {1}{x} \ln (1+x) - 1 \Big)

     = \lim_{x \to 0} \frac {\ln(1+x)-x}{x^{2}}

    which is the indeterminate form  0 \over 0

    applying L'Hospital's rule

     = \lim_{x \to 0} \frac {\frac{1}{1+x}-1}{2x}

    applying L'Hospital's rule a second time

     = \lim_{x \to 0} \frac {\frac{-1}{(1+x)^{2}}}{2} = -\frac {1}{2}


    so the limit of the original function is  e^{-\frac{1}{2}}
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