Is there any book where I can find theory on the limit superior and limit inferior, plus a nice deal of excercises in the same spirit as Spivak's type of excercises?
If it is exercises that you're looking for:
The table of contents may be accessed from the links to AMS page here.
Randomly Chosen Problems from Vol. I:
Randomly Chosen Problems from Vol. II:
Random Page from Vol. III: (Courtesy: the user t.b.)
P.S. (The Indian Editions costs Rs. 800 each volume. Looking hard enough will get you these books from somewhere...)
The first 2 volumes of Kaczor/Nowak have a large number of such exercises scattered throughout. Also, McShane's classic book on Lebesgue integration gives a well written and leisurely treatment. In the U.S., at least, you can find McShane's text in most every college and university library.
Kaczor/Nowak, Problems in Mathematical Analysis, Volumes 1 & 2, American Mathematical Society.
Edward James McShane, Integration, Princeton Mathematical Series, 1944, viii + 394 pages. [My copy is the 1974 8th printing.]
(added next day) After looking at McShane's book, I agree with the comments I wrote yesterday. It gives a very well written discussion of $\limsup$ and $\liminf$ for functions in Article 6 (pp. 26-38). Also, Article 7 (pp. 38-44) is nicely done. Below are 3 additional books that you might find useful (in light of your comment "Spivak's type of excercises"), but which don't specifically devote all that much to $\limsup$ and $\liminf$ for functions (at least, not together in one section).
Ralph Philip Boas, A Primer of Real Functions, 4th edition prepared by Harold Philip Boas, Mathematical Association of America, 1996, xiv + 305 pages.
Andrew Michael Bruckner, Judith Brostoff Bruckner, and Brian S. Thomson, Elementary Real Analysis, Prentice-Hall, 2001, xvi + 677 + 58 pages. [The final 58 pages consist of appendixes: (A) Background, (B) Hints for Selected Exercises, (C) Subject Index.]
Edgar Terome Townsend, Functions of Real Variables, Henry Holt and Company, 1928, xii + 405 pages.
Finally, more advanced readers of this thread may be interested in the following paper, which gives (on pp. 429-430) some relations involving mixed iterations of the operations $\limsup$ and $\liminf$ of a function at a point.
Robert Palmer Dilworth, The normal completion of the lattice of continuous functions, Transactions of the American Mathematical Society 68 (1950), 427-438.
Well,any good analysis text will have a discussion of it,Peter. I personally like the discussion in Angus Taylor's General Theory Of Functions And Integration, which is in Dover and has a lot of excellent exercises. But really,any good analysis text will do.