Confusion regarding the domain of a function Let $f(x)=x^2$ and $g(x)=x$. What is the domain of $\frac{f(x)}{g(x)}$?
Evaluating $\frac{f(x)}{g(x)}$ gives us $x$. Does that mean that its domain is all real numbers?
If we evaluate the function at $x=0$, $\frac{f(0)}{g(0)}$, then $g(0)$ will gives us zero. Does that mean zero is not in the domain of $\frac{f(x)}{g(x)}$?
From what I understand, the domain of $x$ is the set of all real numbers, but the domain of $\frac{f(x)}{g(x)}$ is the set of all real numbers except zero. Am I right?
Edit: $f(x)=2x^2$. Sorry. I forgot to add the two. To make it less confusing, I'm just gonna remove the "2" in $2x$

Evaluating $\frac{f(x)}{g(x)}$ gives $x$ whenever $x\neq0$. So, the domain of $\frac{f(x)}{g(x)}$ is $\mathbb{R}\setminus\{0\}$ (I am assuming that the domains of $f$ and $g$ are both equal to $\mathbb R$.)
In the case of polynomial functions, it is a common convention to extend the domain of $\frac{f(x)}{g(x)}$ to those points $a\in\mathbb R$ such that the limit $\lim_{x\to a}\frac{f(x)}{g(x)}$ exists, and to define $\frac{f(a)}{g(a)}$ as that limit. But that's a convention and it should be clearly stated.
