$f|_A(x)\rightarrow 0$
I know that "|" means "or" or "such that"
I'm not sure what the sub scripted set A means, I'm pretty sure the arrow pointing to zero means that the expression on the left of the arrow approaches zero.
What I think it means is a function such that the set A are the inputs for x, when inputted into the function the function approaches zero.
Yes, you have the right idea. Given any function $f: X \rightarrow Y$, and for any subset $A \subseteq X$, you can define/induce a function $f|_A(x)$, which is $f$ restricted to $A$. It is as if you ignore the rest of $X$, and treat $A$ as the domain of $f$.