I am having a difficult time understanding big-O notation for the growth of functions. My textbook says the following.
Example 2 shows that $7x^2$ is $O(x^3)$. Is it also true that $x^3$ is $O(7x^2)$?
Solution: To detemine whether $x^3$ is $O(7x^2)$, we need to determine whether witnesses $C$ and $k$ exist, so that $x^3\leq C(7x^2)$ whenever $x>k$. We will show that no such witnesses exist using a proof by contradiction.
If $C$ and $k$ are witnesses, the inequality $x^3\leq C(7x^2)$ holds for all $x>k$. Observe that the inequality $x^3\leq C(7x^2)$ is equivalent to the inequality $x\leq 7C$, which follows by dividing both sides by the positive quantity $x^2$. However, no matter what $C$ is, it is not the case that $x\leq 7C$ for all $x>k$, no matter what $k$ is, because $x$ can be made arbitrarily large. It follows that no witnesses $C$ and $k$ exist for this proposed big-$O$ relationship. Hence, $x^3$ is not $O(7x^2)$. $\quad\Box$
I don't understand the part that says, ".... no matter what $C$ is because $x$ can be arbitrary large." Making $x$ large, can't we also assign larger number for $C$?
What are the purpose of $C$ and $k$?
Are there any rules regarding assigning values to them?