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Math Help - limit question

  1. #1
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    limit question

    lim_{x→0+} (1+3x)^(1/x) = eł

    How do I fill in the blanks? In other words, how do I show that they are equal?

    Thank you!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: limit question

    If L=\lim_{x\to a}f(x)^{g(x)} presents an indetermination of the form 1^{\infty} then, by a well known result L=e^{\lambda} being \lambda=\lim_{x\to a}(f(x)-1)g(x) .
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  3. #3
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    Re: limit question

    Thank you! I'm pretty sure I recognize those results by using them in differential equations.
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  4. #4
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    Re: limit question

    Quote Originally Posted by CountingPenguins View Post
    lim_{x→0+} (1+3x)^(1/x) = eł

    How do I fill in the blanks? In other words, how do I show that they are equal?

    Thank you!
    A well known definition of e is \lim_{t \to +\infty} \left(1 + \frac{1}{t}\right)^t.

    So make the substitution 3x = \frac{1}{t} in your limit ....
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