1) Zeno's oxymoronic fleet, stationary arrow: One of the earliest infinity paradoxes, of course, is the flying arrow of Zeno which can't possibly be moving since it takes a finite amount of time to move half the distance AB from one point A to another C and again a finite time for half of that, the length AB/2, ad infinitum, and, since the sum of an infinite number of finite times must be infinite, the arrow could never arrive at C. The resolution comes with the introduction of the concept of the convergerce of an infinite sum, which the paradox points to. (Note Kalai's answer here.)
2) The two-equal-one paradox: Take a horizontal line segment L of length one, form an equilateral triangle from that, and then collapse the triangle by pushing its apex down to L, forming two triangles. Iterate on the apex of each of the two triangles, and then the next four triangles, ad infinitum, and we arrive at the apparent merging of a curve of length two at each iteration to a curve of length one. Resolution and lesson? You'd better impose smoothness, or differentialbility on the collapsing curve. (See this HSE question pointed out to me by David Renfro and "In Search of Infinity" by Vilenkin.) .
What other apparent paradoxes involving going to a geometric limit have been historically instructive and how, i.e., what can they teach us?
I was thinking of explicitly excluding Berkeley's ghosts, but that led to omega-epsilon gymnastics and eventually to Robinson's nonstandard analysis, I believe.
(I believe there are some very sophistcated topological paradoxes of a similar nature, but can't recall their names or where I read of them.)
Note that this excludes primarily set-theoretic or logical paradoxes such as Russell's Barbershop or the Hilbert/Cantor Hotel paradoxes although they are fascinating and important. Bertrand's probability paradox might suit the bill though since it relies on geometrical constructs.