Big $O$, little $o$ Notation So I've gone through the typical undergraduate math sequence (two semesters of real analysis, two semesters of abstract algebra, some measure theory, but I haven't taken discrete math) and in various posts online, I keep on seeing things such as 
$$O(x^2) $$
which I've never encountered in my formal education. What in the world does the big $O$ notation mean, to someone who's completely new to the notation, so that I understand how to use it? (It seems to me like a way to write "some polynomial stuff.")
I've also encountered "little" $o$ notation in studying for actuarial exams.
Also, why do we care about $O$ and $o$? 

Big $O$ notation:
$f(n)=O(g(n))$: 
$\exists \text{  constant } c>0 \text{ and } n_0 \geq 1 \text{ such that, for } n \geq n_0: \\ 0 \leq f(n) \leq c g(n)$
$$$$
Little $o$ notation:
$f(n)=o(g(n))$:
$\text{ if for each constant } c>0 \text{ there is } n_0 \geq 1 \text{ such that } \\ 0 \leq f(n) < c g(n) \ \forall n \geq n_0$
In other words $\displaystyle{\lim_{n \rightarrow \infty} \frac{f(n)}{g(n)}=0}$ 
