I've stumbled upon two different definitions of asymptotic equivalence (very similar but not quite the same): given two functions $f$ and $g$ with $g\neq 0$ in a neighborhood of $x_0$ we say that they are asymptotically equivalent at $x_0$ if: $$\text{DEF 1: }\lim_{x\rightarrow x_0}\frac{f(x)}{g(x)}=1 \qquad \qquad \text{DEF 2: }\lim_{x\rightarrow x_0} \frac{f(x)}{g(x)}=\ell \in \mathbb{R}\setminus\{0\}$$ They don't seem to be equivalent, since for example $2x \sim_2 x$ but $2x \not\sim_1 x$. Personally I feel that the second one is preferable since is more general (and intuitively it makes sense that $2x$ and $x$ are asimptotically equivalent), but I found more often the first one.
Furthermore, I am interested in a specific property of asymptotic equivalence, that is the integration of the equivalence (meaning that if $f \sim g$ then $\int f \sim \int g$). I found some posts that say that this property is satisfied (for example this) and some posts that say the opposite (this). It seems from what I've read that with def 2 this property is satisfied, while with def 1 not necessarily (for example, the counterexample made in the comments of the second post that I linked).
So the problem is that I'm a bit confused by these (apparently, maybe) discordant informations, and I'm trying to put some order and clarification in all this. Any enlightenment?