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I am looking for further examples for two functions $f,g$ such that $f \neq O(g) \land f \neq \Omega(g)$. I have found the old thread Is there a function thats not in Big O and not in Big Omega? where this also discussed, however I wanted to know whether there are also more "basic" functions which fulfill this as opposed to functions with multiple exponentials and piecewise defined functions?

For example if I set $f=sin (x)$ and $g=x\cdot sin(x)²$ would this fulfill the above mentioned property?

M-xyz1
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    Yes, $\sin x$ and $x\sin^2x$ are sound examples as $x\to\infty$. Some authors like to restrict the discussion to monotone functions, though, which is why the author of the answer you linked was stressing for increasing functions. –  Jun 02 '20 at 01:17
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    By the way, one answer mentions that $x\sin x\notin O(1)$ and $1\notin O(x\sin x)$ as $x\to\infty$. –  Jun 02 '20 at 01:24
  • @Gae.S.: Thank you – M-xyz1 Jun 05 '20 at 18:44

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