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Math Help - a limit

  1. #1
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    a limit

    hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !
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  2. #2
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    Quote Originally Posted by dannee View Post
    hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !
    rewrite as ...

    \frac{\ln\left(1+\frac{1}{x}\right)}{\frac{1}{x}}

    ... and use L'Hopital's rule
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    great , thanks !
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    Quote Originally Posted by dannee View Post
    hello everyone, i need help to find the limit of this function: x*ln(1+(1/x)) when x going to infinity. can anyone explain please how should i deal with this one? thanks !
    lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x = ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1
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  5. #5
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    Quote Originally Posted by xalk View Post
    lim_{x\to\infty}xln(1+\frac{1}{x}) = lim_{x\to\infty}ln(1+\frac{1}{x})^x = ln(lim_{x\to\infty}(1+\frac{1}{x})^x) = lne=1
    i don't understand what you did here at all


    lim as x approaches 0 of ln(1+x)^(1/x) = lne = 1
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  6. #6
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    Quote Originally Posted by Noxide View Post
    i don't understand what you did here at all


    lim as x approaches 0 of ln(1+x)^(1/x) = lne = 1
    it's a definition ...

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

    also ...

    \lim_{n \to \infty} \left(1+ \frac{1}{n}\right)^n = e
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