Big-oh and Small-oh Notation: Ratio.

Question

Is it possible to say something about the order of ratios like $O(1/n^2)/o(1)$ or $O(1/n^2)/O(1/\sqrt{n})$?

Answer 1

The $O(f(n))$ and similar notations are sets of functions having a certain behavior as $n\to\infty$. We have $O(f(n))=\{g(n):|g(n)|\le Cf(n){\rm\,for\,\, large\,}n\}$, where $C$ denotes some constant. With this definition, $0\in O(f(n))$ for pretty much any $f(n)$, so the expressions you've written don't make sense.

Answer 2

You have the property if $f \in O(f_1), g \in O(f_2)$, $fg \in O(f_1 f_2)$.

$O(1/n^2)/o(1)$ can be anything. For example, assume you had the numerator being $\Theta(1/n^2)$ and the denominator being $c/n$. Then, this would go to zero as $1/n$. However, if the denominator went to zero exponentially quickly (i.e. the denominator was something like $e^{-c n}$ for some $c>0$), then this would go infinity exponentially quickly.