So I'm studying big $O$ notation right now and am working through a problem and got $O(x^{-10})$ and I'm just wondering if it's possible to even have a term with $O(x^{-n})$ because I've never come across it any of the examples we've gone through in class.
Thanks!
As suggested by the comments, the sensibleness of such an outcome depends on context. As in the comments, this would not be a sensible answer to a question of run-time on input size $x$. It is also not so likely to be a helpful answer if $f(x)=O(x^{-1})$ as $|x|\to 0$. But it might be a sensible answer as part of an asymptotic expansion of a function as $|x|\to \infty$. For example, with $f(x)=1/(1+x^2)$, as $|x|\to\infty$ $|f(x)|=O(x^{-2})$.