Today was another good day. During my study time, I did exercises from the appendix that covers the rigorous definition of limits. I also read a bit of supplemental material on epsilon-delta proofs and part of the next section of the textbook, on limit laws.
Again, the exercises I did today didn’t lend themselves to including in a blog post. They would be routine for math enthusiasts and confusing for others. Faithful readers will have to content themselves with knowing that things are going well.
And faithful readers will rejoice with you. As, I assume, will unfaithful readers, just not as faithfully.
You know, limits are interesting. Calculus books start with limits, and they really are the logical basis on which the subject rests, but they’re by far the most recent part of calculus. By the time limits were being developed in the mid-19th century by Cauchy and by your and my academic ancestor Weierstrass, people had been using derivatives and integrals for 300 years. If you count your guy Archimedes, then people had been using integrals for 2000 years. Something like integration is by far the oldest part of calculus.
The oldest calculus text in Earlham’s library, from the mid-19th century, is interesting to look at partly because it’s a calculus text written before there were limits. The core idea then wasn’t limits, but infinitesimals, which is why they called it the infinitesimal calculus. Early in the book, they define infinitesimals as quantities that you know can’t be 0, but that are too small to measure. They give a few examples, but my favorite is the angular measure of a star. Obviously it can’t be 0, but no telescope of the time could image a star. What’s $dy/dx$? Well, $dy$ and $dx$ are little infinitesimal quantities, and $dy/dx$ is a fraction. Great stuff!
Thank you for this comment. I really enjoyed it. I seem to remember some context in my own study of calculus in which $dy/dx$ was treated as a fraction, but I don’t know what it was.