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Math Help - Finding Limits using a Taylor Series

  1. #1
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    Finding Limits using a Taylor Series

    I need to find :
    <br />
\lim_{x \rightarrow \infty} x-x^{2}ln(1+\frac{1}{x})

    Only I have to use a Taylor Series here, and I mustn't use L'Hopital's rule.

    Thank you very much
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by adam63 View Post
    I need to find :
    <br />
\lim_{x \rightarrow \infty} x-x^{2}ln(1+\frac{1}{x})

    Only I have to use a Taylor Series here, and I mustn't use L'Hopital's rule.

    Thank you very much
    Put u=1/x, then you want:

    \lim_{u \to 0} \left[ \frac{1}{u}-\frac{1}{u^2}\ln(1+u) \right]

    Now expand \ln(1+u) as a Taylor series about u=0 as far as the term in u^3 simplify and evaluate the limit.

    CB
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  3. #3
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    Thank you very much

    BTW - Somebody posted a very practical reply, that said I need to call f(x)=ln(1+x), and use taylor there, then put (1/x) instead of (x), and muliplying it by x^2 and deducting it from x will give me the exact answer. This way seems very nice and 'to the point', but I wonder why the one who wrote it decided to delete it :O ...
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  4. #4
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    That way, of expanding ln(1+x), then putting (1/x) instead of (x), and multiply it by x^2 and all that is a proper proof, isn't it?
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