It reminds me of the preface to Elementary Calculus   an excellent text that I discovered years after learning the subject using the limit:
> The calculus was originally developed using the intuitive concept of an infinitesimal, or an infinitely small number. But for the past one hundred years infinitesimals have been banished from the calculus course for reasons of mathematical rigor. Students have had to learn the subject without the original intuition. This calculus book is based on the work of Abraham Robinson, who in 1960 found a way to make infinitesimals rigorous. While the traditional course begins with the difficult limit concept, this course begins with the more easily understood infinitesimals.
You might want to read the chapter of Calculus Made Easy entitled Epilogue and Apologue ;)
>Thirdly, among the dreadful things they will say about “So Easy” is this: that there is an utter failure on the part of the author to demonstrate with rigid and satisfactory completeness the validity of sundry methods which he has presented in simple fashion, and has even dared to use in solving problems! But why should he not? You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require general rigid demonstrations to be expounded to beginners in the calculus.
You're not wrong, of course, but neither is Silvanus P. Thompson.
Any interesting property of this logic and model is that all functions are infinitely differentiable.
Other introductions are An Invitation to Smooth Infinitesimal Analysis by John L. Bell and Synthetic Differential Geometry by Michael Shulman.
I remember a good portion of both classes finding the concept confusing.
EDIT: It seems like the top-level comment by user woah also finds it confusing.
I remember having to compute the limit of 2x+1 as x goes to 3 by writing it as the limit of 2x + the limit of 1, and then saying the first limit is 2 times the limit of x, and so on. Utterly boring, especially when the teacher couldn't even answer us when we asked "if we already know we'll get 7 by setting x to 3, why do we have to write all this?"
The whole thing almost put me off math.
There's a time and place to introduce properties of limits, but the first month of calculus 1 isn't it.
I think calculus should be taught like "here's a set of tools to help you rewrite your equations to make them output the slope of their graph".
Then, afterwards, you can get into what makes it all tick. Sort of like teaching a student to write "hello world" before diving into the low level mechanics of a compiler.
I feel the exact opposite way. The method you're describing is how I was taught calculus in high school, that is, with a strong focus on practicing mechanical manipulation of formulas rather than building deep understanding. And I just disconnected from that, I got no mental reward from procedurally finding the derivative or the integral of a function for the 100th time. Lack of motivation meant lack of focus, which in turn made my performance drop, and at the time I assumed I probably just wasn't as good at mathematics as I thought I was. That lasted until I got mathematics classes in college, where many of my mathematics courses used a more first-principles-based approach, and I performed well again.
> Sort of like teaching a student to write "hello world" before diving into the low level mechanics of a compiler.
I found Structure and Interpretation of Computer Programs (https://mitpress.mit.edu/sites/default/files/sicp/index.html) the most enlightening intro to programming book I ever read. Its approach is also strongly first-principles-based.
I'm not writing this comment to say that you're wrong; the method you describe may very well work well for you, and many others with you. Rather, I'm using the opportunity to spread an opinion that I feel quite strongly about: the best way to learn a topic is completely student-dependent and there is no such thing as the one true way in which a topic should be taught. The most powerful motivator is always intrinsic motivation, and different minds are motivated by different challenges. Unfortunately it is hard to bring into practice in a batch-processing based education system.
And we derive a few formulas along the way. integrals of x and derivatives of x2 (x^2, depending on your programming language) pop up all over the place.
I do wonder though -- for heavily skill-based endeavors (say swimming, sushi-cheffing, curling, etc.), the best way to learn is to do. First principles/mental model approaches tend not work so well in these domains, at least not isolation.
The best approach is probably a blend: learn by doing and understanding first principles.
I think it would be quite difficult to do in a reasonable amount of time, but using history as a course outline would be a cool way to explain the motivations/context behind the principles being discussed. That might better illustrate how math research works by explaining the gaps/interesting questions that existed prior to certain ideas and how those ideas evolved and lead to other gaps/questions.
The type of math courses I’ve gotten a lot out of already kind of did that, they just usually used more of a condensed, logical narrative than a historically based one.
It's still really important to work through problems mechanically (often with only partial understanding) if only to become familiar with patterns. Whether this exploration is done via a CAS or by hand, acquiring this mechanical familiarity is important if one is to go deep. Math is often not merely read but done.
Take something like automatic differentiation (AD) for instance. It's just the chain-rule in principle right?
One could try to study the principles of AD in textbooks/journal pubs and then try to implement it in code. Chances are one is unlikely get it right the first few times because anyone who's ever tried to implement anything from numerical computation papers knows that published papers regularly omit important implementation details (unless they also publish code). There's so much heuristic/unintuitive behavior  in the world of numerical computation that first principles alone is rarely enough to get you a decent implementation in code.
On the other hand, if one mechanically works through AD derivations for a bunch of functions and observing the patterns that emerge (without necessarily grasping all the principles at first), one begins to notice things, like difficulties posed by corner cases like non-smooth/discontinuous functions, non-n'th differentiable functions, etc. A CAS could be used for this exploration (manually doesn't necessarily literally by hand) but it's so important to actually try stuff out and really grasp how mathematical objects work in practice. Once this contextual understanding is acquired, going back to AD journal pubs and reading about the principles is likely going to make way more sense because one has seen the behavior "in the field".
I do think the pure mechanical repetition is often underrated as a autodidactic mechanism. (again I don't mean literally by hand, but rather "doing" instead of just "reading and thinking").
 non numerical comp folks are often surprised to hear this because they think everything's super deterministic -- it's not. Tiny random perturbations in matrix can change things wildly. Take naive Gaussian elimination -- the ordering of the rows of equivalent linear systems can vastly affect solution efficiency and numerical stability, but one can't appreciate that unless one understand how GE works mechanically and has worked through examples.
It reminds me of when I was learning antenna theory and propagation. I was looking for something that started at step one. The best videos I found on YouTube were a series created in the 1950's by the Royal Canadian Air Force.
"Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can."
It made me believe I could really learn calculus by myself at age 12. After reading the first few chapters, nope. Turns out calculus is built on a foundation of algebra and geometry which I did not have until I was 16. Once I did have the foundation though, calculus became easy and mostly mechanical. But to a 12 year old, this book overpromised and underdelivered.
I also found the exposition a little wordy and tedious, like it was written to explain calculus to an English major. It's the sort of book that one appreciates in retrospect after knowing the subject, but not while learning it. I discovered the most effective way for learning math is actually not by reading but to mechanically work through problems to cultivate intuitions and to develop a pattern matching schema. In doing so, one gains confidence that one can actually solve problems. Once this confidence is achieved, going back and delving into the underlying principles becomes so much more contextualized and rewarding.
Learning math by reading books like this is like learning to ride a bike by reading about it. You try to understand all the principles, but when you need to deploy them, you find yourself unable to execute. Much better to do it the other way around.
EDIT: but I want to soften that by saying that I appreciate not everyone learns this way. It's just I've seen too many struggle with math even though they've read the textbook countless times... when a simple change in stategy would yield much better results.
A book with an easy approach really needs to make the chapters super easy to restart and spiral through concepts. Stories and characters didn’t hurt to have too though.
Aside from Axler, I wish someone picked up Halmos text and rendered it in a more modern way. It's a really good one, a bit more advanced, but the text is so cramped it's unpleasant to follow.
As Euclid said, "there is no royal road to geometry."
It goes well with 3blue1brown's calculus series on YouTube, which gives more of a visual intuition for the concepts.
(4) A piece of string 30 inches long has its two ends joined together and is stretched by 3 pegs so as to form a triangle. What is the largest triangular area that can be enclosed by the string?
Just replace inches with your linear measure of choice, mm, rods, poles, chains, stadia, light years, or anything else; it makes no difference to the problem. Thompson could have said 'units of length' instead of 'inches'; but the whole purpose of the book is to build on easy notions that Thompson believed everyone could manage and perhaps even be familiar with already, so he used imperial units that were familiar to every handyman of the time instead of metric which would be familiar to far fewer and hence distracting.
Also don't forget that there were several distinct 'metric' systems in the past before most of the world settled on SI (MKS, CGS, etc.)
15. With the development of purely peaceful space research, the President-Director-General of the Societe Anonyme pour l'Exploitation Financiere de la Physique Purement Theorique, Commander of the Legion of Honneur, had been able to conclude some profitable contracts and has bought a country estate of 200 hectares in Basse-Normandie at 8,000 francs per hectare. Inspired by this example, a plumber employed by the company, earning 800 francs a month, decided to invest a tenth of his wages in savings bonds at 4% interest per annum. How many years will he have to work before he is in a position to buy an estate of 200 hectares in Basse-Normandie and live out the rest of his life in peace and comfort? (You should take account of compound
interest, but ignore the effects of possible devaluations of the currency.)
16. According to Le Monde of 21st July 1954, the annual expenditure on the war in Indo-China is given by the following table (the unit is 1,000 million old francs):
Use this example to verify the fundamental properties of addition (associativity and commutativity).
17. In November 1954, there were in Algeria 1,230,000 Europeans and 8,300,000 native-born Algerians. At the same date, the University of Algiers had 4,548 European and 557
Algerian students. Calculate the ratio between the chances of a European and those of an Algerian of receiving higher education.
And insert some ads: Please click here to buy some protractors for 25% off.