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For all positive functions $f$ and $g$ of the real variable $x$, let $\sim$ be a relation defined by

$f \sim g$ if and only if $\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 1$

Then if $f \sim g$, we have for example, that $f^2 \sim g^2, \sqrt{f} \sim \sqrt{g}, f+g \sim 2g$, but we do NOT have that $e^f \sim e^g$. What other operations preserve, or do not preserve the asymptotic relation $\sim$?

  • 1
    There are some things to point out in your examples. Each demonstrates *functional composition* i.e. functions $h$ such that $$f\sim g \implies h\circ f\sim h\circ g,$$ but these are only a subclass of operators preserving the $\sim$ relation (or are they?). If you put extra restrictions on the functions, your class of operators may expand fortuitously - l'Hôpital's suggests we could include differentiation if each of $f$ and $g$ are asymptotically either $0$ or $\infty$ (and are differentiable for sufficiently large $x$). Finally, your last example puts $g$ on both sides of the relation...2011-10-11
  • 0
    You may find more complete answer or/and reference in my [post](http://math.stackexchange.com/a/1684489/102814).2016-03-05

2 Answers 2