I especially liked his points on examination, what we call the "orals" in Physics. There, the (in his words, defenseless) student is up in front of a board, while professors throw problems at them. I had a few good questions on mine, and one which was ... poorly specified. I remember thinking I simply had to crank on that problem, showing my thinking processes to try to answer the question. At the end, the prof nodded, pointed to something before my conclusion, and said "that was as far as I got."
I remember feeling relieved yet angry. Just smiled, nodded, thanked him for the question, and moved on.
Talk about an imbalanced power dynamic.
I would have loved to have a simple 100 known-but-non-trivial questions like this. You avoid the lottery of having to remember some particular detail (say some integral that appears in some derivation that an adhoc question might contain), but you don't avoid having to actually know how the theory works, because it's a little bit too hard to memorize.
Admittedly you might still get stuck on a trivial step but at least you've had a chance to go over the questions, and it might be relatively fresh.
Edit: someone posted a link showing that one can generalize this result for g, f and their inverses. Interestingly, after reading up on it, the point Arnold is trying to make is that modern students have a poor grasp of infinitesimals.
I would expect a journal to not use a different font integral sign, however. Or rather, I would expect the "house style" to include font choices for integral signs as well. I'm not aware of any journal that uses this type of integral now, but I suppose I should also note that I mostly read papers from the arxiv now anyway.
(So the limit is essentially (x - x)/(x - x) )
To make this rigorous do as you say about expanding the series.
Edit: Up to beyond first order actually*
I agree with the intuition, but this is not so simple. A limit such as yours can still be anything, can't it? It depends on the higher-order terms. For example, the limit ((x+5x^2)-(x+2x^2))/((x+7x^2)-(x+6x^2)) is also "essentially" (x-x)/(x-x), but it turns out to be 3, not 1. In the given example, there are cancellations of even higher order terms, that you have to check that they cancel correctly. Following the links on the answer by admissionguy I found two nice arguments for the computation, an algebraic and a geometric one.
Question 2 in particular, I'd be curious if it can get there without guidance.
[EDIT]: Answer is yes: https://www.wolframalpha.com/input/?i=Limit%5B%28Sin%5BTan%5...
It seems a bit different to the other questions (but is question 1, so might be an "easy" starter.)