TITLE: how to calculate the limit of an integral?
QUESTION [6 upvotes]: 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!

REPLY [3 votes]: 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.