Sometimes, when we take limits, especially for roots and ratio tests, we define lim of sup(a_n). Is this only because when we take the limit of sup we don't have to worry about the existence of the limit? Then what about lim inf ?
I'm sure this could be conceptually pretty easy questions, but then again, they're just thrown at the reader in some books about anaysis, so I thought I should probably ask...
It is not "the limit of sup", but the lim-sup or the limit supremum, not "of". Why we take that is clear from the context of convergence of series and, of course, because it works that way.
Now, many times the test is applied in cases where the limit, just like this, exists finitely and, thus, both limit supremum and limit infimum exist and are equal to the limit, so we don't have to worry about supremum/infimum and stuff. Unless one studies mathematics, there is a good chance that some easier version of these tests is given/proved/used, without having to worry about the more general one.