Question about $o$, $O$ and $\sim$ Let $f$ and $g$ be functions defined on real line and let $g(x)\ne 0$ from some sufficiently large $x$.
Please let me know if this is correct:


*

*$f\sim g \iff \lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$

*$f=O(g) \iff \lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=c$, where $c$ is some positive constant

*$f=o(g) \iff \lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$


Not quite. You should have


*

*$f\sim g \iff \lim_{x\rightarrow\infty}\left|\frac{f(x)}{g(x)}\right|=1$

*$f=\Theta(g) \iff \lim_{x\rightarrow\infty}\left|\frac{f(x)}{g(x)}\right|=c$, where $c$ is some positive constant

*$f=O(g) \iff \lim_{x\rightarrow\infty}\left|\frac{f(x)}{g(x)}\right|=c$, where $c$ is some non-negative constant

*$f=o(g) \iff \lim_{x\rightarrow\infty}\left|\frac{f(x)}{g(x)}\right|=0$


Also, it is not strictly necessary that we take these limits as $x \to \infty$, but that part is context dependent.
