Precise intuition about limits and infinitesimals

[Sleek]

I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals.

For example, I know that `0.999... = 1`, where both the LHS and RHS are the same numbers with different representations (there's a proof that uses limits). How can I formally understand this? What branch of mathematics can I start exploring, and what are the best resources to do it?

Also, what are some good resources on the history of limits and the technical understanding of something "tending to infinity but not infinity?"

On my background: I've done math heavily in my undergraduate studies and calculus is not a problem. However, I'm trying to get deep into making things that I've learned intuitive, and not just resort to manipulating symbols without complete understanding.

Please let me know if my question is a bit vague, I'll be happy to add more details.

 

[jedishrfu]

THere is a sequence of videos on Youtube by 3blue1brown called the Essence of Calculus where he discusses the notions behind limits in a very visual way. They are excellent videos that deserve to be watched by anyone interested in understanding the foundations of Calculus.

https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

 

[fresh_42]

There is an excellent book from Jean Dieudonné about the mathematical history between 1700 and 1900. The numbers are from its title, the content isn't as strict at its borders. Unfortunately I don't know of an English version, but it's really a good source, and as I find, exciting.

 

[member 587159]

It is possible to make infinitesimals rigorous. This is called "non-standard analysis", and is treated in the book "Non standard analysis" by Robinson.

 

[Stephen Tashi]

How can I formally understand this? What branch of mathematics can I start exploring, and what are the best resources to do it?

Also, what are some good resources on the history of limits and the technical understanding of something "tending to infinity but not infinity?"

A formal and technical understanding of limits is a different goal than understanding the early history of these concepts. The early history of the concept of limits is primarily useful in intuitively understanding limits. To understand the formal concept of limits, you need to understand the "game" of modern mathematics. This involves understanding the use of formal logic and logical quantifiers - and the outlook that definitions mean what they say as opposed to being descriptions of things that already exist.

It would be nice if mathematical topics could be studied in a gentle way, starting from their historic roots and proceeding to modern treatments. However, this is a time consuming way to learn things and most expositions that take this approach assume you already appreciate modern mathematical formalism.