The picture is a question from the book Intro to Electrodynamics by Griffiths. In question as you can see we want to find potential due to an infinite strip maintained at constant potential in the given region (slot).
My question is that the fourth boundary condition they used is ($V\to0$ as $x\to\infty$) which I think is wrong . Because when we deal with infinite charge distribution (like infinite line of charge or infinite sheet of charge) the convention that potential zero at infinity fails. For infinite charge distribution potential at infinity blows up. And hence we cannot use ($V\to0$ as $x\to\infty$) condition. So what is wrong with my argument here?