What is the useful criterion in practice to judge $f(x)\in O(g(x))$ as $x\to c$?

Question

Let $c\in\Bbb R$. Does $\displaystyle \lim_{x\to c}\left\vert \frac{f(x)}{g(x)}\right\vert$ exists if and only if $f(x)\in O(g(x))$ as $x\to c$? I think $\Rightarrow$ direction is correct, however not sure the opposite direction. Anyway, I want to ask if there's a quick/useful criterion to judge whether $f(x)\in O(g(x))$ as $x\to c$, instead of directing checking the definition.

Answer 1

For the first question, let $f(x)=x\sin(1/x)$ and $g(x)=x$, then $|f(x)|\leq|x|=|g(x)|$, but $f(x)/g(x)=\sin(1/x)$ has no limit as $x\rightarrow 0$.