In asymptotics: why is $o(f(x) g(x))=o(f(x)) O(g(x))$ and $o(f(x) g(x))=f(x) o(g(x))$?

Question

Using the small-$o$ and large-$O$ notation in asymptotical analysis, I am trying to understanding why these two relations hold: $$o(f(x)\ g(x))=o(f(x))\ O(g(x)),\ \ \ \ \ (x\rightarrow0)$$ $$o(f(x)\ g(x))=f(x)\ o(g(x)),\ \ \ \ \ (x\rightarrow0)$$

Any help in explaining this would be appreciated.

Answer 1

Using definition.Now we know $o(f(x)), x\to 0$.So,

$\lim\limits_{x\to 0}\frac{o(f(x))}{f(x)}=0=\lim\limits_{x\to 0}\frac{g(x)o(f(x))}{g(x)f(x)}.$By definition, $g(x)o(f(x))=o(g(x)f(x)).$

Because $\frac{O(g(x))}{g(x)}$ is bounded,so o(f(x))O(g(x))/f(x) close to $0$ when $x\to 0$. So,$o(f(x)g(x))=o(f(x))O(g(x))$.

Actually,$x\to x_0\in\mathbb{R}$, it’s also right.