I can't understand the difference between these two definitions
A function $f$ is said to be asymptotic to a function $g$ as $x \to +\infty$ if $lim_{x\to +\infty} f(x)-g(x)=0$
That means $f(x)=g(x)+o(1)$ as $x\to +\infty$
A function $f$ is said to be equivalent to a function $g$ as $x \to c \in \mathbb{R} \cup \big\{+\infty,-\infty \big \}$ if $lim_{x\to c} \frac{f(x)}{g(x)}=1$. In that case we write $f \sim g$ as $x\to c$
This means $f(x)=g(x)+o(g)$ as $x\to c$
What are the relation and the difference between the two?
I think that this relation:
$f=o(g)$ as $x\to c$ $\implies$ $f=o(1)$ as $x\to c$
holds when $g \sim x^{n}$ whith $n>1$ as $x\to c$
So if $f$ is asymptotic to $g$ then is it also equivalent to it?
Secondly I found the first definition just for $x \to +\infty$ but is it valid also for $x \to c \in \mathbb{R}$ ?
Thanks in advice for your help