Week 10 - Limits and Non-Polynomial Functions

I get the feeling that logical notation has a hidden contempt for calculus and limits.  Prof. Heap is always saying that limits don't quite express the true nature of a function as some variable approaches infinity. He's of the opinion that numbers can never really approach infinity but instead they just get "very large" or "move farther way" from some value such as the origin. We also have our own expressions for limits that are always 3 or 4 times as long as the calculus version and I find that it adds an unnecessary complexity to a proof.

Take for example the limit as n approaches infinity for  2ⁿ / n².  By calculus we can easily show that this approaches infinity.  Then by inspection we can quickly say that a function of 2ⁿ is guaranteed to surpass n² as n gets large thus n² is in O(2ⁿ) and 2ⁿ is in Ω(n²).  The proof using calculus, would probably be about 2 lines.  But the proof using logical notation and the structures of this course is approximately 5 to 6 times as long which really impedes our understanding of the underlying concept of the relationship between two functions. The meaning of big-O and big-Ω are hidden behind this excessive notation.

I know limits themselves have a similar precise definition, but they also have a higher level notation that quickly expresses the long term behavior of a function. Personally, I've had difficulty with limit proofs since the beginning of the course. To help I've reviewed several of the limit chapters from my old calculus text book and I've also tried drawing diagrams showing the delta and epsilon boundaries.  But when the function changes it can easily throw you off and put you right back at square one where you need to reinterpret the strict definition of the limit. I plan on spending a greater portion of time with limits of various types such as ones that approach infinity, zero and some other value and for various functions as well.  Hopefully there are some good examples in the course notes that I can practice on.

Here is someone who shares the same confusion with limit proofs.
http://slogcsc165.blogspot.ca/2014/11/week-11-recap-from-previous-weeks.html?showComment=1417548152625#c1153904731624398479