In this question on one of the sister sites of m.s.e., the following problem is exhibited:
From the point $(\cos\theta,\sin\theta)$ on the unit circle, drop a perpendicular to the $x$-axis, and consider the region bounded by that perpendicular, the $x$-axis, and the arc from $(1,0)$ back to $(\cos\theta,\sin\theta)$. Ask what fraction of the area of that region is between the chord from $(1,0)$ to $(\cos\theta,\sin\theta)$ and the arc from $(1,0)$ to $(\cos\theta,\sin\theta)$, and then take the limit of that fraction as $\theta\downarrow0.$
The only solution I know of involves either a routine application of L'Hopital's rule or a routine use of power series, and L'Hopital's rule must be iterated three times.
It is somewhat impressive because such a simply stated geometry problem can defy intuition so thoroughly and yet L'Hopital conquers it so neatly.
But in a comment under the question linked above, someone asked whether there is a "practical reason" why someone would want that limiting ratio.
I will construe that in the following way, and make it my present question:
Is there a reason originating outside of this problem – say in geometry or geodesy or astronomy or engineering or theoretical physics or statistics-applied-to-demography or whatever – for an interest in that limiting ratio?