My name is Dan Lima and my grandfather Antonio Barcelos Lima came from Terceira. He settled in California and died in Los Banos. I have no information on your ancesters, but I do know that there are tons of Lima's that came from Terceira. Somehow, I'm sure that we're all related, but I haven't found the missing pieces yet.
All that I can tell you is my lineage: Dan Lima (me) (1977-current), Joe Lima (1955-current) , Antonio Barcelos Lima (1925-1995), Antonio Lima (1888-1970), Antonio Lima (1866-?), Manuel Lima (?-?).
If any of these men fall into your lineage, please tell me. That's all that I know so far.