Vladimir Arnold on formal thinking

The solution has no limit, of course, because the phase curve is a closed oval.

However, the statement of the question assumed (erroneously) that the solution does have a limit. If it were true that the solution approaches a limit, then the argument given by these students would be completely correct.

The students should have recognized that the solution does not approach a limit, but rather is periodic.

Not myself being a professional mathematician, I have had a lot of professors express similar frustrated sentiments to my lectures.

Notwithstanding the fact that in the anecdote Arnold provides his students are merely responding to his own error in an examination setting (what else would he expect them to do but try to solve the problem given with the tools they'd been taught?), it sounds like he's making a distinction between a person understanding the formal components that make up a proof, and a person having a coherent intuition of the subject as a whole that would guide them to the formal components that make up a proof.

If you subscribe to Church's Thesis then you could explore Arnold's sentiments by the following extreme example:

Consider the programming language Malbolge, which was invented to be pathological. While programming is possible, and you could create a running program on your computer by copying someone else's code, or even by stringing together working functions that people have created, you could not afterwards claim to understand Malbolge like you might claim to understand Python. Nor could someone who modifies a working 'Hello World' program in Malbolge to print 'Goodbye World' claim to understand how Malgolbe interprets the function as a whole.

That's my interpretation of the passage anyway. Soft answer for a soft question, hope it was helpful.

Surely,the students' answers were logically valid,but not logically sound? More experience with similar faulty questions would help them develop the habit of checking all initial assumptions in any given questions, and through this 'personal empiricism' gain intuition. Simply put, trust nothing, check everything.