Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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
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

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.

share|cite|improve this answer
I have solve that problem. The limit is $+\infty$. – jenny Oct 13 '11 at 2:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.