на самом краю

на самом краю — on the very brink
Слово краю, в котором падеже? Почему не - на самом крае?

Actually it is the Locative case. In Russian for most words it coincides with the Prepositional case (therefore usually it is not mentioned in elementary textbooks). But for some words it is not the same as Prepositional: в лесу, на снегу, на краю, на двери, some others...

Спасибо, it-ogo.