My professor says that we use $f(x) = O(g(x))$ to mean that $g(x)$ is negligible compared to $f(x)$. Im kind of confused here. The definition I found online is
$$ f(x) = O(g(x)) \iff \exists C >0, \; \; st \; |f(x)| < C |g(x)| \; as \; x \to 0$$
With this definition I can see that for example $x < e^x $ for $x \to 0$ and so $x = O(e^x)$ how is $e^x$ negligible compared to $x$? Am I misunderstanding the concept completely?
Personally, I would use the word "negligible" for Little-Oh. The way I like to think of it is $$f(x)=O(g(x)) \Leftrightarrow \lim_{x\to\infty}\frac{f(x)}{g(x)}<\infty$$ and $$f(x)=o(g(x)) \Leftrightarrow \lim_{x\to\infty}\frac{f(x)}{g(x)}=0$$ So Big-Oh means "$f$ doesn't grow too much bigger than $g$" and Little-Oh means "$f$ becomes negligible compared to $g$"