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Math Help - lim n->infinity

  1. #1
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    lim n->infinity

    Hi!,

    I have a problem solving this.. I am half way through it but somehow cannot finish it. Any help is really appreciated

    lim [ (n ^ ln n)/ (a ^ n)]
    n->00

    Thanks in advance!
    Cheers!
    Ramya
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  3. #3
    Senior Member DeMath's Avatar
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    Quote Originally Posted by Ramya View Post
    Hi!,

    I have a problem solving this.. I am half way through it but somehow cannot finish it. Any help is really appreciated

    lim [ (n ^ ln n)/ (a ^ n)]
    n->00

    Thanks in advance!
    Cheers!
    Ramya
    Consider the case 1 < a < + \infty

    \mathop {\lim }\limits_{n \to \infty } \frac{{{n^{\ln n}}}}<br />
{{{a^n}}} = \mathop {\lim }\limits_{n \to \infty } \exp \ln \frac{{{n^{\ln n}}}}{{{a^n}}} = \mathop {\lim }\limits_{n \to \infty } \exp \left\{ {\ln {n^{\ln n}} - \ln {a^n}} \right\} =

    = \mathop {\lim }\limits_{n \to \infty } \exp \left\{ {{{\ln }^2}n - n\ln a} \right\} = \mathop {\lim }\limits_{n \to \infty } \exp \frac{{{{\ln }^4}n - {n^2}{{\ln }^2}a}}{{{{\ln }^2}n + n\ln a}} =

    = \mathop {\lim }\limits_{n \to \infty } \exp \frac{{{{\left( {\frac{{\ln n}}{{\sqrt n }}} \right)}^4} - {{\ln }^2}a}}{{{{\left( {\frac{{\ln n}}{n}} \right)}^2} + \frac{{\ln a}}{n}}} = \exp \frac{{{{\left( {\mathop{\lim }\limits_{n \to \infty } \frac{{\ln n}}{{\sqrt n }}} \right)}^4} -{{\ln }^2}a}}{{{{\left( {\mathop {\lim }\limits_{n \to \infty } \frac{{\ln n}}<br />
{n}} \right)}^2}}}=

    = \exp \frac{{{{\left( {\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {\ln n} \right)}^\prime }}}{{{{\left( {\sqrt n } \right)}^\prime }}}} \right)}^4} - {{\ln }^2}a}}{{{{\left( {\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( {\ln n} \right)}^\prime }}}{{{{\left( n \right)}^\prime }}}} \right)}^2}}} = \exp \frac{{{{\left( { - \frac{1}{2}\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\sqrt n }}} \right)}^4} - {{\ln }^2}a}}{{{{\left( {\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}} \right)}^2}}} =

    = \exp \frac{{0 - {{\ln }^2}a}}{0} = \exp \left\{ { - \infty } \right\} = 0.
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  4. #4
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    Thanks a ton!
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