When you input 0/x on W|A, it claims, with no restrictions on $x$, that $0/x=0$. While of course it claims $0/0$ is indeterminate.
I'd like it to be more rigorous in this regard. It is typical for W|A to assume an unspecified letter $x$ is a complex variable. In this case, as always, it should assume $x$ is any complex variable, but it assumes $x\neq 0$, without pointing out the assumption. I'm honestly bothered by this.
The Generic and Non-Generic Cases tutorial in the Wolfram Language documentation center explains the motivation for this behavior.