It's a stretch to say that people after Newton did not know what a limit was. It's just that they did not have the formal definition of one.
Newton invented limits. Only he did not have a fundamental basis for it. Hence, he saw it is a dirty trick that happened to yield correct and useful results.
The formalization of limits came later when the intuitive use of a limit started failing. If I recall, this was with the Weierstrass function.
I think we agree though. You can do most maths just by assuming you can do things with the rational numbers. It takes some really weird stuff before you need to properly define the 'real-numbers'.
Newton invented limits. Only he did not have a fundamental basis for it. Hence, he saw it is a dirty trick that happened to yield correct and useful results. The formalization of limits came later when the intuitive use of a limit started failing. If I recall, this was with the Weierstrass function.
I think we agree though. You can do most maths just by assuming you can do things with the rational numbers. It takes some really weird stuff before you need to properly define the 'real-numbers'.