Aren't these time consuming ways?

Perhaps. This is something to test. If portability is not a priority, system "cat source >> dest" is probably better with unixes. OP wants to avoid memory issue:

OP: Finally, I would want to know the right and best way to append a to another file with out consuming memory.

So I presume read/write in buffer corresponding to blocksize is a good choice if we can know the good one. With files of such size (terrabytes) there is a question: do I really have to append files? :)