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Math Help - Find the limit of (1+x)^(3/x)

  1. #1
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    Find the limit of (1+x)^(3/x)

    I'm trying to find the limit of (1+x)^(3/x) as x approaches 0. I've tried a million different ways to change the power to something where I can simplify 3/x by multiplying (1+x)^(3/x) by [(1+x)^y]/[(1+x)^y], where y is sinx or x^2/x or something else so that I can add it to 3/x and get something that I wouldn't have to divide by zero. .... I hope that doesn't sound too confusing.... Oh, and I need to know how to do this without a calculator, so if you could explain that to, that would be so totally amazing!!!

    Thanks!
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  2. #2
    Member anonimnystefy's Avatar
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    Re: Fidn the limit of (1+x)^(3/x)

    use the fact that the limit of (1+x)^(1/x)=e.

    you should get e^3.
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  3. #3
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    Re: Find the limit of (1+x)^(3/x)

    Quote Originally Posted by NothingButSmiles View Post
    I'm trying to find the limit of (1+x)^(3/x) as x approaches 0. I've tried a million different ways to change the power to something where I can simplify 3/x by multiplying (1+x)^(3/x) by [(1+x)^y]/[(1+x)^y], where y is sinx or x^2/x or something else so that I can add it to 3/x and get something that I wouldn't have to divide by zero. .... I hope that doesn't sound too confusing.... Oh, and I need to know how to do this without a calculator, so if you could explain that to, that would be so totally amazing!!!

    Thanks!
    An alternative way is to let t = \dfrac{1}{x} \ \Leftrightarrow \ x = \dfrac{1}{t}. Therefore as x \rightarrow 0^+ \text{  then  } t \rightarrow \infty


    This will enable you to rewrite your limit as \lim_{t \to \infty} \left(1+\dfrac{1}{t}\right)^{3t} = \lim_{t \to \infty} \left( \left(1+\dfrac{1}{t}\right)^{t}\right)^3


    That last limit is a classic way of defining e
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