I would like gain a better understanding of the ancient Greek notion of limit.
In particular, I am asking for some good references (books especially) that explain well, for example, why the ancients thought that an asymptotic curve meets its asymptote "at infinity.'' I presume that this would have also included numerical sequences as well. I would furthrmore like to discover any connections between this and the idea that parallel lines ``meet at infinity"---which I understand may have motivated Euclid's 5th postulate.
If anyone can offer any good insight to the ancient notion of limit---that will be appreciated; but I am especially looking for good references that expound well on topics related to the ancient notion of limit and the their idea of two things ``meeting at infinity.'' (no wikipedia please)
Thank you.
So according to the modern notation and concepts these ideas of 'meeting at infinity' are meant to be taken as conceptual gimmicks and rules of thumb which are not precisely true, but rather explain in laymen's terms a more sophisticated structure or process which preserves some of these notions as limits are taken, and which are shown through an epsilon delta arguments as per the usual route.
Regarding the notions of the ancient Greeks as they thought them is more useful from an historical perspective. The ancient Greeks generally had a notion and definition of limits which gave rise to many paradoxes which we find easy to resolve. Consider for example Zeno's paradoxes, particularly the paradox of Achilles and the Tortoise. You will have to look at another answer to understand the historical perspective in depth. I'm not a math historian.
It may therefore be more interesting to turn your attention toward modern rigorous mathematical structures which formalize certain notions of behavior at infinity, meeting at infinity, or the notion of the infinitesimal.
Concerning parallel lines in the plane meeting at infinity you may wish to consider the Riemann Sphere representation of the complex plane where this notion can be made precise in a formal way.
Concerning an asymptotic curve meeting it's asymptote at infinity there are several different extensions of the standard real numbers where something like this can be made formal.
One of the most extensive attempts to formalize these notions is called Non-Standard Analysis. Non-Standard Analysis attempts to formalize the earlier intuitive proofs and concepts introduced by Gottfried Wilhelm Leibniz by extending the set of reals with an additional set of hyperreals representing infinite and infinitesimal (infinitely small but non-zero) quantities and defining a set of principles which describe their interactions.