I'm studying for my discrete math class and I don't fully understand how to proof how a function is not a big O for certain questions. I understand that you have to assume that it is big O and proof by contradiction.
Please explain how you attacked these questions step by step.
1) $1$ is not $O(\frac{1}{x})$
2) $e^x$ is not $O(x^5)$
1) Suppose $1=O(1/x)$. Then there exists a constant $C$ such that for all large $x$, we have $1 \le C/x$. However, for all $x>C$ we have $C/x < 1$, a contradiction.
2) Suppose $e^x=O(x^5)$. Then there exists a constant $C$ such that for all large $x$ we have $e^x \le Cx^5$. However, $\frac{e^x}{x^5} \to \infty$ as $x \to \infty$, a contradiction. [As Peter suggests, use L'Hopital's rule multiple times, or expand the Taylor series for $e^x$.]