Little-o notation: limit in the definition

We usually point out the limit of $x$ in the text, such as "$f=o(g)$ as $x\to0$" for the first case and "$f=o(g)$ as $x\to\infty$" for the second case.

Yes. For example, $f(x)=x$ and $g(x)=1$.

I'm gonna cite another way to define it without limit of $\frac{f(x)}{g(x)}$, maybe this will result easier:

Let be $f : U \longmapsto \mathbb{R},x_{0}$ accumulation point of $f$. We say that $$f = o(g(x)) \hspace{0.2cm} x \to x_{0}$$

If exists $\epsilon : U \longmapsto \mathbb{R} :$

$$f(x) = \epsilon(x) \cdot g(x), \hspace{0.3cm} \lim\limits_{x \to x_{0}} \epsilon(x) = 0$$