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## Homework Statement

True Or False

if f(x) continuous in [tex][a,\infty][/tex] and [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex]

Then [tex]\int_a^\infty f(x)dx [/tex] converge

## Homework Equations

Anything from calc 1 and 2

## The Attempt at a Solution

Actually I'm really stuck..

My main motive is to try and show that if [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex] there is no way that Cauchy test can fail,

I assume Cauchy test fail and that [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex] and try to show that there is no option for this to happen.

First I tried on f(x) that can be positive and negative .. didn't made it, Then I tried on f(x) that is only positive also didn't made it..

Maybe its false, But I didn't found any function that does not converge and also [tex]\lim_{x\to\infty}\int_x^{2x}f(t)dt = 0 [/tex]

Thank you.