Is it true that if $$\lim_{x\to +\infty} f(x)=+\infty$$ then $f $ can not be integrable at the neighborhood of $+\infty$, hence the improper integral $\int_0^{+\infty}{f(x)dx}$ does not exist?
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2yes, that is true. – Ittay Weiss Jun 20 '14 at 20:38
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Can you please tell me the main reason behind this? – palio Jun 20 '14 at 20:39
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@palio Check your definition of integrability. What's one of the main conditions required on $f$? – Git Gud Jun 20 '14 at 20:40
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@GitGud the limit : $\lim_{X\to +\infty}\int_0^Xf(x)dx$ must be finite. – palio Jun 20 '14 at 20:44
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4Hint: If $f(x)>M$ for $x>a$ then $\int_a^b f(x),dx>(b-a)M$ when $b>a$. – Harald Hanche-Olsen Jun 20 '14 at 20:46