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June 14th, 2017, 08:45 AM   #1
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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
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June 14th, 2017, 08:52 AM   #2
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Consider what happens if it tends to a non-zero value.
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June 15th, 2017, 06:07 PM   #3
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Use fact integral of 1/x diverges as x becomes infinite.
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June 16th, 2017, 04:26 AM   #4
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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.
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June 17th, 2017, 04:45 AM   #5
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Quote:
Originally Posted by Micromike View Post
Where does the 1/x come from?
Also as x tends to inf, 1/x tends to 0
Yes, it does. But what is the integral of 1/x? What is $\displaystyle \int_1^\infty \frac{1}{x}dx$?

"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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