ca.analysis and odes - A question regarding a claim of V. I. Arnold

Here is a problem which I heard Arnold give in an ODE lecture when I was an undergrad. Arnold indeed talked about Barrow, Newton and Hooke that day, and about how modern mathematicians can not calculate quickly but for Barrow this would be a one-minute exercise. He then dared anybody in the audience to do it in 10 minutes and offered immediate monetary reward, which was not collected. I admit that it took me more than 10 minutes to do this by computing Taylor series.

This is consistent with what Angelo is describing. But for all I know, this could have been a lucky guess on Faltings' part, even though he is well known to be very quick and razor sharp.

The problem was to find the limit

$$ lim_{xto 0} frac
{ sin(tan x) - tan(sin x) }
{ arcsin(arctan x) - arctan(arcsin x) }
$$

The answer is the same for
$$ lim_{xto 0} frac
{ f(x) - g(x) }
{ g^{-1}(x) - f^{-1}(x) }
$$
for any two analytic around 0 functions $f,g$ with $f(0)=g(0)=0$ and $f'(0)=g'(0)=1$, which you can easily prove by looking at the power expansions of $f$ and $f^{-1}$ or, in the case of Barrow, by looking at the graph.

End of Apr 8 2010 edit

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Beg of Apr 9 2012 edit

Here is a computation for the inverse functions. Suppose
$$
f(x) = x + a_2 x^2 + a_3 x^3 + dots
quad text{and} quad
f^{-1}(x) = x + A_2 x^2 + A_3 x^3 + dots
$$

Computing recursively, one sees that for $nge2$ one has
$$ A_n = -a_n + P_n(a_2, dotsc, a_{n-1} ) $$
for some universal polynomial $P_n$.

Now, let
$$
g(x) = x + b_2 x^2 + b_3 x^3 + dots
quad text{and} quad
g^{-1}(x) = x + B_2 x^2 + B_3 x^3 + dots
$$

and suppose that $b_i=a_i$ for $ile n-1$ but $b_nne a_n$. Then by induction one has $B_i=A_i$ for $ile n-1$, $A_n=-a_n+ P_n(a_2,dotsc,a_{n-1})$ and $B_n=-b_n+ P_n(a_2,dotsc,a_{n-1})$.

Thus, the power expansion for $f(x)-g(x)$ starts with $(a_n-b_n)x^n$, and the power expansion for $g^{-1}(x)-f^{-1}(x)$ starts with $(B_n-A_n)x^n = (a_n-b_n)x^n$. So the limit is 1.