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Math Help - find the limit

  1. #1
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    Question find the limit

    lim ->∞ 1/x+1

    x^{1/x} as x approaches infinity? I know that limit of (1+1/x)^{x} = e (as x-> infty), and limit of (1+x)^{1/x} = e (as x-> 0), but this is slightly different.

    Any help is appreciated
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  2. #2
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    Re: find the limit

    Quote Originally Posted by LAPOSH42 View Post
    lim ->∞ 1/x+1

    x^{1/x} as x approaches infinity? I know that limit of (1+1/x)^{x} = e (as x-> infty), and limit of (1+x)^{1/x} = e (as x-> 0), but this is slightly different.

    Any help is appreciated
    Note that x^{1/x} = e^{\frac{\ln(x)}{x}} so I suggest you apply l'Hospital's theorem to \frac{\ln(x)}{x}.
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  3. #3
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    Re: find the limit

    Quote Originally Posted by mr fantastic View Post
    Note that x^{1/x} = e^{\frac{\ln(x)}{x}} so I suggest you apply l'Hospital's theorem to \frac{\ln(x)}{x}.
    And also note that due to the continuity of the exponential function, that \displaystyle \lim_{x \to a}e^{f(x)} = e^{\lim_{x \to a}f(x)}
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