Limit of length of curves vs length of limit of curves

Question

3Blue1Brown clip shows a false proof on $\pi=4$, as we fold a square onto a circle, with the frame

What exactly is the distinction between $lim(len(c))$ and $len(lim(c))$? I heard vaguely about continuity and Euclidean vs Taxicab distance but these don't explain it to me. Following the integral definition of arc length and it involving the curve's derivative, I do see the argument on the velocity vector, but can't make a connection to this. I'd expect $lim$ and $len$ to commute.

What would be the exact mathematical expression for each? (Or for a close enough alternative example, showing lack of commutativity with full mathematical expressions)

https://math.stackexchange.com/questions/2679894/what-is-the-proof-behind-lim-fgx-f-lim-gx — insipidintegrator, Jul 14 '22 at 14:21
why do you expect $\lim$ and $\text{len}$ (typically defined using an integral, which itself is a limiting process) to commute? THe fact that two limits do not commute is not a flaw in analysis, in fact it is one of the more difficult and exciting features of analysis, and gives us a whole bunch of stuff to study. There are usually conditions under which such commutations are possible, but it's not always possible, and for the circle and $\pi=4$ troll proof, commuting is not valid. — peek-a-boo, Jul 14 '22 at 14:21
See The staircase paradox, or why $\pi\ne4$ AND What is the history of staircase or =4 paradox? AND Lower semicontinuity of length of graph: $L(g)\le\liminf_{n\to\infty}L(f_n)$ AND these other questions. — Dave L. Renfro, Jul 14 '22 at 14:25
Thanks all for the references, I'll try to look through sometime. It'd help if the list was narrower to pinpoint the commutativity issue. — OverLordGoldDragon, Jul 14 '22 at 15:15
@peek-a-boo I'm certainly not calling it a flaw, it's indeed intriguing. I just don't see how Euclidean and Taxicab distances come to disagree in this particular limit, and can't think of an example where distance computation commutation fails. — OverLordGoldDragon, Jul 14 '22 at 15:16
The example $\frac{1}{n}\sin(n^2 x)$ is discussed in: Nancy Edwards, An instance of intuition and lengths of limiting curves, The Pentagon 31 #1 (Fall 1971), 22-25 & 45. The following discusses a similar example, along with when such a paradox doesn't occur (roughly, for some $a \leq b$ we have $a \leq f'_n(x) \leq b$ for all $n$ and $x),$: John Anthony Terilla, On the convergence of arclength, Journal of Undergraduate Mathematics 23 (1991), pp. 25-27. — Dave L. Renfro, Jul 15 '22 at 07:08
A rather detailed historical discussion of curve length is given in: Gilbert E. Traub, The Development of the Mathematical Analysis of Curve Length from Archimedes to Lebesgue, Ph.D. dissertation (under Kenneth Philip Goldberg), New York University, 1983, ix + 455 pages (degree awarded in 1984). — Dave L. Renfro, Jul 15 '22 at 07:20
Hi @OverLordGoldDragon I found a similar question to yours: https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4 — insipidintegrator, Jul 21 '22 at 07:52
Re: duplicates - I'm sure somewhere there lies the answer to my question, but I'm asking on SE to get an SE answer and save time, not to have to dig through extended text. — OverLordGoldDragon, Jul 23 '22 at 00:18

Limit of length of curves vs length of limit of curves

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