Maybe this will come across as an unpopular opinion, but I wanted to share how I feel sometimes as a math student.
At the moment, I'm taking multivariable calculus for math majors, and it's probably the first time in my career that I've had to rigorously prove stuff and think over my homework (as opposed to just plugging and chugging as we're often encouraged to do in introductory calculus).
Having said that, it sometimes seems like I'm still having to resort to a similar studying routine when solving exercises. For example, when going over epsilon-delta limit proofs, I'm only guessing ways to work around them that we've already tried in class. Sometimes a particular procedure works fine for a function, sometimes it doesn't and you need a novel way to prove the limit that you hadn't thought about. But why is it that such things happen to succeed for some particular functions and not for others, I have no clue about.
As another example, when evaluating when a function is continuous and taking the limit for some path, I have to hope that the algebra will work out somehow to get what I want (again, having no idea why some do and some don't). And the same seems to happen to me when calculating directional derivatives and the like.
It's not just limits though, when evaluating boundaries for particular sequences I also felt similarly, as I mainly tried to make algebra magically prove that the sequence was decreasing (which I already knew from plugging in a few numbers).
I think it's not that I don't know the theory, but that most of the exercise applications consist of guessing ways to make algebra do its magic. And that bothers me a lot, considering the rigor involved in the theoretical area of the subject.
All in all, I just don't get what's the purpose of having these kinds of problem exercises. I would've expected to have homework that played around a bit more with theorems and stuff. Not just every problem consisting of a random function that is very difficult to analyze and requires "artistic" solutions. I don't want to sound like a stereotypical high-school kid, but why should finding whether $$\frac{e^{x^{2}+y^{3}}}{xy-x+y^{2}}\ $$ is continuous at (0,0) even matter? It's pretty unlikely I'll ever find such an ugly function in real life, and if I ever do, I will have already forgotten how to go about proving it has a limit at the origin (if it even does, I honestly didn't solve it).
I'm sorry for the rant. To be frank, I only decided to write this after I flunked my exam today. I'll probably change my mind in a few days, but until then, I thought it might be healthy to express my frustration. It's partly my fault, as I didn't do all the problem sets, but I lack the motivation to do them for the reasons I described above. I feel like even if I had done them, I would've failed regardless, as they don't seem to build on your intuition (maybe I'm wrong here, almost everybody praises practicing math as a way to understand the subject better).
Anyway, thanks for taking the time to read my long post! I would really appreciate any kind of answer, be it agreeing or disagreeing with me.