Studying Mathematics Thread

[Jehanne]

I have read that the modern-day definition of the limit in calculus was near the last quarter of the 19th-century, with the last book (other than the University of Wisconsin's foray back in time) on infinitesimals being around 1915.

Cauchy did his stuff in the early part of the 19th. By mid-century, the epsilon-delta definition was standard among mathematicians (if not among others).

Some final touches, apparently, a century ago:

Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed variable quantities, infinitesimals, and limits and defined continuity of y = f ( x ) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y in his 1821 book Cours d'analyse, while (Grabiner 1983) claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).

The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).

I lament the fact, though, that many modern calculus texts no longer have a separate, concluding chapter on ODEs. A step backward in my opinion.