Unsuspected dangers of extrapolating from truncated analyses

Differential calculus is central to the physical sciences. Analyses of phenomena described by Maxwell's equations, Navier's equation, the Navier-Stokes equations, or the Korteweg-deVries equation, etc., all need differential calculus. Limits are essential to differential calculus. Differentiation of a function f(x) about a point x0 as a casual perusal of calculus textbooks as well as Newton's notebooks will suffice to show,3 has a geometric interpretation in which the spread (xb - xa) of the range xa ≤ x0 ≤ xb is made increasingly small until it almost vanishes. Assuming that f(x) is continuous, this limiting process yields its derivative at x0. Or, does it?

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