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If $$\lim_{x\to\infty} f(x) = \infty$$ what can be said about the rate at which $$\int_1^\infty f(x) \,dx$$ approaches infinity if $f(x) \geq 1$ for all values of $x$?

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All that can be said is that $$\frac{\int_1^x f(y)dy}{x}\rightarrow \infty.$$ No better lower bound can be given, and nothing can be said about the rate at which this goes to infinity since nothing is given about $f$. Indeed, you can construct $f$ so that this ratio goes to infinity as slowly, or as quickly as desired.

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Is this all the information given? If so, not much.

$f(x)$ could be $|x| + 1$ or $e^x + 1$. These are two pretty different functions, in terms of the rate at which their integrals grow. You would need more information to say much of anything at all.

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So nothing can be said about it? Is there any lower bound? – user1825464 Jan 24 '13 at 5:05
@user1825464: Lower bound on how quickly it diverges? It wouldn't seem so since you can modify B.D.'s examples to $(|x|+1)^{c}$ for $c>0$ and it will diverge slowly for small $c$, or $(e^x+1)^{c_0}$ which can cause the function to diverge quickly. – Clayton Jan 24 '13 at 5:09
The integral can even go to infinity more slowly than $f(x)$. Use for example $2xe^{x^2}$. – André Nicolas Jan 24 '13 at 6:03

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