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Math Help - Problem with limit

  1. #1
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    Problem with limit

    Any help to the following?

     \lim_{x \rightarrow 0} \left(\frac{ln (x+1) }{x} \right)^{\frac{2}{X}}
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  2. #2
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    Quote Originally Posted by atlantis84 View Post
    Any help to the following?

     \lim_{x \rightarrow 0} \left(\frac{ln (x+1) }{x} \right)^{\frac{2}{X}}
    One possible approach:
    1. Substitute the following Maclaurin series: \ln (x + 1) = x - \frac{x^2}{2} + \frac{x^3}{3} - .... .

    2. Simplify the result.

    3. Consider \lim_{x \rightarrow 0} e^{\ln \left( 1 - \frac{x}{2} + \frac{x^2}{3} - .... \right)^{2/x}} = \lim_{x \rightarrow 0} e^{2 \ln \left( 1 - \frac{x}{2} + \frac{x^2}{3} - .... \right)/x} .

    4. Consider the limit of the exponent using l'Hopital's rule.


    Alternatively, there may be a clever substitution that can first be made.
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