June 14th, 2017, 08:45 AM  #1 
Newbie Joined: Jun 2017 From: Yaoundecameroon Posts: 2 Thanks: 0  Functions.
We are given a function which is continuous, positive decreasing and integrable on [1,$\infty$[. We are required to show that as x tends to infinity, xf(x) tends to 0. How do we proceed?
Last edited by Micromike; June 14th, 2017 at 08:55 AM. Reason: An omission 
June 14th, 2017, 08:52 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,442 Thanks: 1462 
Consider what happens if it tends to a nonzero value.

June 15th, 2017, 06:07 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,416 Thanks: 558 
Use fact integral of 1/x diverges as x becomes infinite.

June 16th, 2017, 04:26 AM  #4 
Newbie Joined: Jun 2017 From: Yaoundecameroon Posts: 2 Thanks: 0 
Where does the 1/x come from? Also as x tends to inf, 1/x tends to 0 Last edited by skipjack; June 17th, 2017 at 06:21 AM. 
June 17th, 2017, 04:45 AM  #5  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,922 Thanks: 785  Quote:
"1/x" is sort of the "border" of "integrable functions". $\displaystyle \frac{1}{x^\alpha}$ is integrable, from 1 to infinity, as long as $\displaystyle \alpha> 1$. Last edited by skipjack; June 17th, 2017 at 06:21 AM.  

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