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Thread: L'Hopital's Rule

  1. #1
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    L'Hopital's Rule

    Could someone help me use L'Hopital's Rule to find the following limit?

    lim (as x approaches 0) (1+(3/x))^x
    Thank you.
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  2. #2
    TD!
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    You need to get an indeterminate form such as 0/0 or inf/inf to apply l'HŰpital.

    $\displaystyle \left( {1 + \frac{3}{x}} \right)^x = e^{\ln \left( {1 + \frac{3}{x}} \right)^x } = e^{x\ln \left( {1 + \frac{3}{x}} \right)} $

    Now we can use the fact that:

    $\displaystyle
    \mathop {\lim }\limits_{x \to 0} e^{x\ln \left( {1 + \frac{3}{x}} \right)} = e^{\mathop {\lim }\limits_{x \to 0} x\ln \left( {1 + \frac{3}{x}} \right)}
    $

    Now you have two possibilities, one is:

    $\displaystyle
    x\ln \left( {1 + \frac{3}{x}} \right) = \frac{{\ln \left( {1 + \frac{3}{x}} \right)}}{{\frac{1}{x}}}
    $
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