How does the method of exhaustion lead to the concept of a limit?

[NoraBD]

I’ve been trying to understand why the method of exhaustion is considered a precursor to calculus, but I keep getting stuck on how it actually bridges to the concept of a limit. When I read about Archimedes using it to find the area of a circle, I follow the steps, yet the leap to a modern limit feels vague.

[Luke66]

I did the Archimedes polygon thing myself, drawing the inscribed triangles inside the circle and then doubling the sides. Each time the area got closer to the circle’s true area, and the gaps shrank in a predictable way. It felt like a pattern you could chase forever, but the leap to calling that a limit never showed up on the page I read.

[Chloe57]

Watching numbers drop the gap, I started to think of it as a sequence that would settle somewhere. Still, I kept worrying I was reading it wrong, like maybe the modern limit is a different animal and I was just restating what’s obvious.

[TimothyS]

Another angle that tripped me up is whether the method of exhaustion is really about a limit at all, or just about filling in the area with more and more careful bits. I kept wondering if the real problem was me trying to map a geometric story to an algebra rule, and whether I was chasing the wrong thing.

[Paisley65]

I tried to teach this to myself with a quick notebook sketch and a few doodled polygons, and the numbers improved a little, then stalled. It felt tangible, but the jump to modern rigor felt stubbornly abstract, like I could see the end of the process but not the path.