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**zzzhhh** Thank you for your reply. I'm reading Rudin's "Principles of Mathematical Analysis", this book introduces infinite limits(P98) just before the definition of derivatives(P103), so I naturally have the question if infinite limit is regarded as "exists" or not. According to your reply, it is not, so I concluded that we have to confine ourselves to traditional real set in dealing with limits of functions in the definition of derivatives, but I'm not sure, that's my second post. I have refered to other books such as "Mathematical Analysis" written by Apostol, but all these books do not explain explicitly whether infinite limit is allowed. Could you please cite some textbooks stating explicitly that, in the definition of derivatives, infinite limit means the limit does not exist? Thanks!

ps: I have not study real analysis yet. I plan to study it after I finish baby Rudin, and to use Rudin's "Real and Complex Analysis" as textbook, is this book good for self-study?