Exploiting  Limit  Definition

.99.. vs 1 a Real Problem

The history of calculus is important to understand, particularly how Newton addressed the concept of 0/0. The symbol of the integral, ∫, is derived from the Latin word "summa," indicating summation. Most people  know that the concept of integration comes from summation. Newton used Δx to calculate derivatives and invented calculus. People like Cauchy attempted to eliminate the need for Δx by introducing the concept of limits. However, they fall short to replace dx(Δx) or prove that it actually zero. Changing name of Δx or delta x  dx didn't solve the problem and just buried the infinitesimal for a century or so . We know that an integral is the sum of dx (Δx) * f'(x)=f(x). The point is that dx (Δx) is essentially an infinitesimal and proof that it cannot be zero. The limit is the heart of calculus and is not well-defined( or poorly defined ) , even though it contradict the notion of Δx being zero. If dx (Δx) is zero, then all integrals would be zero( sum of f'(x)*0=0, which is not the case. https://www.youtube.com/shorts/In2msBtAZto.

We understand that assuming certain things can simplify our lives, like thinking the Earth is flat when we go shopping, play basketball, or drive to another city using a map. However, ease of use isn't proof that the Earth is flat; similarly, assuming delta x is zero in calculus makes calculations easier and seems logical until proven otherwise. 

 Most people today think "rigor" means correct and proven facts. Rigor actually refers to a set of carefully examined and  accepted rules, which are not necessarily correct but are assumed to be true. Occasionally, if these rules are found to be incorrect or contradictory, they are either discarded or revised. So, when we refer to the "rigorous definition" of calculus, it simply means we have decided to accept it as correct. That does not constitute a proof, nor does it guarantee correctness.