TITLE: Formally proving that a function is $O(x^n)$
QUESTION [5 upvotes]: Say I have a function 
\begin{equation*}
f(x) = ax^3 + bx^2 + cx + d,\text{ where }a > 0.
\end{equation*}
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $k, x_0 > 0$ exist, such that $0 \le f(x) \le kg(x)$ for all $x > x_0$.
My question is, what are appropriate values for $k$ and $x_0$? It's easy enough to find ones that apply (say $k = a + b + c + d$). By the formal definition, all I have to do is show that these numbers exist, so does it actually matter which numbers I use? For some value of $x$, $k$ could be anywhere from $1$ to $a + b + c + d + ... $. From my understanding, it doesn't matter what numbers I pick as long as they 'work', but is this right? It seems too easy.
Thanks

REPLY [6 votes]: HINT $\quad\rm  ax^3 + bx^2 + cx + d\  \le \ (|a|+|b|+|c|+|d|)\ x^3 \ $  for $\rm\ x > 1$