This question has arleady been asked, but I'm not satisfied with answers given there.
I've been struggling for some time with understand the true meaning and inuition behind of $o$ and $O$ notation.
Definitions that I've been given are following:
$D1$. $f=O(g)$ when $x \to x_0$, if there is neighborhood $U$ of $x_0$ and constant $c>0$, such that: $$|f(x)| \leq c|g(x)|$$ when $x \in U$.
$D2$. $f=o(g)$ when $x \to x_0$, if there is neighborhood $U$ of $x_0$, such that: $$f(x)=\epsilon(x)g(x)$$ when $x \in U$, where $\epsilon$ is infinitesimal function when $x \to x_0$.
I don't really understand the $O$ notation.
For when we are talking about $o$ notation, I think that this several cases are the most important ones, and rest is not really applicable (?).
If $g$ is infinitesimal too, then so is $f$, and we can say that $f$ tends to zero faster than $g$.
But if $g,f \to +-\infty$, then $g$ tends to $+-\infty$ faster then $f$.
But what if $g,f$ are not from the cases above ? What if they don't have limit when $x \to x_0$ ? And does it have to be explained in terms of $tends to$ ? Someone sometime said to me, not really precise,that it has sommeting to do with $growth$, but I can't see how to apply that to this separate cases that I listed.