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Math Help - limit with integral in denominator

  1. #1
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    limit with integral in denominator

    What is \lim_{t->\infty}\frac{x^{t}}{\int_{0}^{\infty}u^{t}e^{-u}du} ?
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  2. #2
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    This is \underset{t\to \infty }{\mathop{\lim }}\,\frac{x^{t}}{t!}. With no information over x, we can't do much.
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    Quote Originally Posted by Krizalid View Post
    This is \underset{t\to \infty }{\mathop{\lim }}\,\frac{x^{t}}{t!}. With no information over x, we can't do much.
    One "cheap" way to do this is to assume there is one specific answer, independent of x. If the answer does not depend on x, take x= 0 and immediately the sequence is 0, 0, 0, 0,...

    If x is not 0, we can at least assume x> 0 since otherwise x^t is not defined for some t. And it is well known that t! dominates any x to the t power. One way of looking at it is this: the numerator and denominator both include t factors. In the numerator, the factor is always x. In the denominator the factors are from 1 up to t so for large enough t, the denominator is larger than the numerator- and as t gets larger the difference gets greater.
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  4. #4
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    Quote Originally Posted by Krizalid View Post
    This is \underset{t\to \infty }{\mathop{\lim }}\,\frac{x^{t}}{t!}. With no information over x, we can't do much.
    Take x>1 or something, why would \int_{0}^{\infty}u^{t}e^{-u}du be equal to t! ?
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  5. #5
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