Big O problem $2^n$ and $n^2$

Question

How can i prove that $$2^n\not \in O(n^2)$$ by formal definition and not using limits?

With:

Answer 1

$2^{n} > n^{2}$ for $ n > 4$ (which can be proved by induction). now apply the definition and prove

Answer 2

It needs to be shown that there do not exist constants $c, n_0 > 0$ such that $2^n \le cn^2$ for all $n \ge n_0$. It suffices to show that the ratio $\frac{2^n}{n^2}$ gets arbitrarily large and therefore can become larger than any $c$ value that is chosen. To prove that $\lim_{n \rightarrow \infty} \frac{2^n}{n^2} = \infty$, apply L'Hospital's rule twice.