# how to calculate the limit of an integral?

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 $(0,1)$. – Did Oct 10 '11 at 20:03
Is this homework? If so, please tag as such. – cardinal Oct 10 '11 at 20:05
Could you please show me the detail? – jenny Oct 10 '11 at 20:25
Could you please show us what you know, what you tried, where you are stuck? Thank you so much! – Did Oct 10 '11 at 20:32
how to deal with the case for $t$ near zero? – jenny Oct 10 '11 at 20:35

## 1 Answer

I would start by calculating, for fixed $x$ and $t$, the value of the supremum in the integrand. (Note: since the integrand is $e^{-(st-1)\log s}$, this is equivalent to finding where $(st-1)\log s$ is smallest on the interval $(x,\infty)$.)

The answer probably depends on the relationship between $t$ and $x$, and so your integral will split up into two integrals - perhaps one will be a function of $t$, while the other will be some constant depending on $x$. Hopefully then you can evaluate the function in parentheses explicitly as a function of $x$, at which point taking the limit will be easier.

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I have solve that problem. The limit is $+\infty$. – jenny Oct 13 '11 at 2:49