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Math Help - Prove that lim x->inf (x ^ (1/x)) = 1

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    Prove that lim x->inf (x ^ (1/x)) = 1

    "Prove that lim x->inf (x ^ (1/x)) = 1."

    I'm stuck on this. I was able to show that the limit exists, is finite, and therefore unique. But showing that it equals 1 is eluding me. If I raise both sides to the xth power, I get x = 1^x, which is clearly wrong. So what do I do to show that the limit is 1? Proof by contradiction? Induction? Any hints on how to proceed?
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    Senior Member jakncoke's Avatar
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    Re: Prove that lim x->inf (x ^ (1/x)) = 1

    do you mean lim_{x \to \infty} inf(x^{1/x}) = 1 ?
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    Re: Prove that lim x->inf (x ^ (1/x)) = 1

    Quote Originally Posted by phys251 View Post
    "Prove that lim x->inf (x ^ (1/x)) = 1.

    You know that if y=x^{f(x)} then y=e^{f(x)\ln(x).

    What is \lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?.
    Thanks from phys251
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    Re: Prove that lim x->inf (x ^ (1/x)) = 1

    Quote Originally Posted by jakncoke View Post
    do you mean lim_{x \to \infty} inf(x^{1/x}) = 1 ?
    Yes.

    Quote Originally Posted by Plato View Post
    You know that if y=x^{f(x)} then y=e^{f(x)\ln(x).

    What is \lim _{x \to \infty } \frac{{\ln (x)}}{x} = ?.
    That limit would be zero. So I would just take e^0=1 and be done?
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    Senior Member jakncoke's Avatar
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    Re: Prove that lim x->inf (x ^ (1/x)) = 1

    --
    Last edited by jakncoke; February 13th 2013 at 01:39 PM.
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    Re: Prove that lim x->inf (x ^ (1/x)) = 1

    Quote Originally Posted by phys251 View Post
    That limit would be zero. So I would just take e^0=1 and be done?
    Yup, it indeed was that simple. Thanks, Plato!
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