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Could you please tell me how to calculate the limit: $$\lim_{x\rightarrow+\infty}\left(\int_0^1\sup_{s>x}\frac{s}{e^{(s\log s)t}}dt\right)$$ Thank you so much!

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    The limit is zero. To show this, a first step is to prove that the value at $t$ of the function to be integrated converges to zero when $x\to+\infty$, for every fixed $t$ in $(0,1)$.2011-10-10
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    Could you please show me the detail?2011-10-10
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    Could you please show us what you know, what you tried, where you are stuck? Thank you so much!2011-10-10
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    how to deal with the case for $t$ near zero?2011-10-10

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