Disproving $\mathcal O$ in a general way

Question

I believe $5+3n^3=\mathcal O(n)$ is false but I really don't know how to disprove it! Can someone provide me a template or technique that can be used when disproving $\mathcal O$ or even $\Omega$?

Answer 1

If $5+3n^3=O(n)$ by the definition of big O there is some constant $c$ for which $$ 5+3n^3\leq cn $$ For any $n$.

Shouldn't be too hard to derive a contradiction here.