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I have not found the complete list of operations that we can perform on asymptotic equivalences using the $\sim$ notation. This equivalence is defined as follows: $f \sim g$ if $\lim_{x \to \infty} f(x)/g(x) = 1$.

What about the composition? Let us suppose that we have $f \sim g \circ h_1$ and $h_1 \sim h_2$. Can we write $f \sim g \circ h_2$?

Thank you.

Dingo13
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1 Answers1

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No, it does not hold in general. Take $g(x)=e^x$, $h_1(x)=x^2+x$ and $h_2(x)=x^2$. Then $h_1\sim h_2$ but $g \circ h_1\not\sim g \circ h_2$: $$\frac{(g \circ h_1)(x)}{(g \circ h_2)(x)}=e^{x^2+x-x^2}=e^x\to+\infty$$ as $x\to+\infty$.

Robert Z
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