My only experience is as a physics TA and teaching X-ray techs, so take this with a grain of salt. I think the author is trying to address a real problem, but he's not working on the right parts.

First, limits are harder than derivatives. Historically, humans figured out the derivative in the late 1600s, but the modern rigorous definition of the limit didn't exist until the 1800s. Slow-walking the definition of the derivative doesn't fix the problem of understanding limits.

The limit of a function f at a point x is defined as the value y, if it exists, such that for all d > 0 there exists an e > 0 such that for all x' in [x - e, x + e] we have |y - f(x')| < d. That's an earful. But for essentially all limits in introductory calculus we evaluate using two rules: the limit of a continuous function f at a point x is f(x), and the squeeze theorem. So my suggestion is to elevate these to the status of axioms. Introducing another number system does not help when students will not do anything nontrivial with it anyway.

The second problem is that "introductory" calculus includes too much material and is consequently pushed too late in the curriculum and seen as a weed-out course. Students spend too much time on "preparation" that doesn't prepare them for calculus. Studying logarithms and trigonometry is orthogonal, so basically all of "precalculus" is not actually pre-calculus. To me a four-year high school curriculum could be written up just fine with two years of algebra and geometry (not separated), one year of calculus and then statistics, which provides an ideal application for the theory of derivatives when you learn regression. But the author has included multivariable calculus and fiddly techniques for taking derivatives of ugly functions into "introductory" calculus. I think this is a step in the wrong direction. Laborious algebra calculations can be moved into an optional methods course for engineering students; we should be ensuring the core ideas are as accessible as possible so that doctors don't write papers about the trapezoid rule anymore:

https://diabetesjournals.org/care/article/17/2/152/17985/A-M...