Did Sauron Really Know Frodo?

[Evenstar1]

I agree with what Nar has just said, partly because that was the impression I had when reading, and partly because I just really like this thought: Sauron didn't know who had the Ring and where he was until the very last second, when Frodo put the Ring on at the edge of Mt. Doom. And forever after, Sauron will have the name and face of Frodo Baggins emblazoned in his brain.

This is totally in-line with Tolkien's eucatastrophe-event, to have the Ultimate Bad Guy realize his imminent demise at the last second, when it is too late, and he realizes that the one who did him in is some shrimpy-halfling who has made it all the way across his barren wasteland and up the side of a mountain unnoticed! (Except for the one part where the clumsy henchmen had caught him but somehow let him escape.) And previously, Sauron hadn't even bothered to differentiate between Hobbit-names and faces (as akhtene pointed out), let alone even really noticed the entire RACE of Hobbits!