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I have question related to asymptotic relation $\sim$. First is the following: if we know that $f(x)\sim g(x)$ as $x$ tends to $\infty$ what we can say about the asymptotic behaviour of $F(x)$ as $x$ tends to $\infty$, where $F(x)=\int\limits_{0}^{x}f(s)ds$ and $f$ is locally integrable positive function? I suspect that primitive function of $g$ must be included in it in some way but I can't see how. I tried to use $\varepsilon-\delta$ technique but I stuck. Thank you for any tips.
Edit: maybe some simple examples could be helpful.
1. $f(x)\sim \frac{1}{\sqrt{x}}$ as $x$ tends to $\infty$. If that means $F(x)\sim \sqrt{x}$ as $x$ tends to $\infty$?
2. $f(x)\sim \frac{1}{x^{2}}$ as $x$ tends to $\infty$. If that means $F(x)\sim \frac{1}{x}$ as $x$ tends to $\infty$?

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    The suggestion in part 2. of the Edit is extremely odd. Unfortunately, the OP quit the page without leaving explanations.2012-07-01

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