§ 327 Practical Mathematics

On 8 April 2010 J. H. S. asked a question regarding a claim of V. I. Arnold: In his "Huygens and Barrow, Newton and Hooke", Arnold mentions a notorious teaser that, in his opinion, modern mathematicians are not capable of solving quickly. Calculate
lim_(x-->0) (sin(tanx)-tan(sinx)) / (arcsin(arctanx)-arctan(arcsinx))

The answers given in
including the accepted one (with 31 upvotes) show that professional research mathematician in fact cannot solve such problem other than in a very ponderous way.

My answer was shorter: "I think Arnold alludes to the idea that is often used in physics lessons, for instance when treating mechanical oscillators. For small x we can put x = sinx = tanx = sintanx = arcsinarctanx etc. This yields the limit 1 immediately." [WM, 29 May 2013]

For professional research mathematicians this sounds too primitive. In fact many don't understand at all was has been said. User Misha, for instance, asked: "This gives you lim_(x-->0) (x-x)/(x-x) = 1. I guess, you are using a system of axioms where 0/0 = 1."

Constantin, a Greek mathematician on sabbatical in Germany, dared to defend my position asking Misha: "Would you disagree that (x-sinx)/(x-sinx) = 1 for every x including the limit? That same holds for tan, arctan and so on? In my opinion Arnold cannot have expected that someone calculates limits. Either you see it - or not."

Few hours later all his contributions were deleted without any announcement and Constantin had been suspended for a month. Later he has been deleted completely.

The impression that the "great research-level logicians of MathOverflow" are not so great in sober mathematical thinking must be avoided by all means.

PS: Of course also my answer has been deleted. Certainly it was not "research- level".

Regards, WM