This might be a very dumb question, but I have googled around and looked in asymptotic analysis reference texts and cannot seem to find what I am looking for. So here we go:
Suppose I know that there exists some $x$ such that $xn \sim n^2$ as $n \to \infty$. Can I conclude that $x \sim n$?
Yes, because equivalence is compatible with multiplication, so $$xn\sim n^2\quad\text{and}\quad \frac1n\sim\frac1n\quad\text{imply}\quad xn\cdot \frac1n=x\sim n^2\frac1n=n.$$