In 'Related notations' chapter:
We write $f(x) = o(g(x))$ for $x \to a$ if and only if for every $C>0$ there exists a positive real number $d$ such that for all $x$ with $|x - a| < d$ we have $|f(x)| < C |g(x)|$.
If $g(x)\neq 0$, this is equivalent to $\lim_{x \to a} f(x)/g(x) = 0$.
How can we prove that these two definitions are equivalent?
You apply the definition of $\lim$ directly, that is for example $\lim f(x)/g(x)=0$ means that for every $C>0$ there exists a $d>0$ such that $|f(x)/g(x)|<C$ whenever $|x-a|<d$, and that $|f(x)/g(x)|<C$ is equivalent to $|f(x)|<C|g(x)|$ (if $g(x)\ne0$). The $d$ (depending on $C$) is the same in both definitions.
If $g$ is not the zero map, note that $$ \bigg| \frac{f(x)}{g(x)}\bigg| < C $$ iff $$ |f(x)| < C|g(x)|. $$
