What order of growth does a ratio of Bigh-Ohs have?

Question

Say that $f(n) = \cal O(n^2)$ and $g(n) = \cal O(n)$.

If $h(n)=f(n)/g(n)$, is it true that $h(n) =\cal O(n)$?

Is it mathematically correct to say that $h(n) = \cal O(n^2)/ O(n) = O(n)$? if not, what would be the correct way to show this?

Answer 1

Let $f(n)= n^2$ and $g(n)=1$. Then $f(n) = \cal O(n^2)$ and $g(n) = \cal O(n)$.

However, $h(n)=f(n)/g(n)=n^2/1 = n^2 \neq \cal{O}(n)$.