Comments on: Why Do We Need Limits and Infinitesimals?

By: Kalid

Kalid — Thu, 11 Aug 2011 08:42:14 +0000

@werterber: Not a silly question at all! In my head, it’s saying “what’s the ratio of width [cos(x)] to distance traveled (x)”.

As our distance traveled goes to 0 (we aren’t moving from the starting point), cos(x) tends towards 1 — we’re pretty much at the same width. So it becomes “1 / 0″ in my head.

By: werterber

werterber — Thu, 11 Aug 2011 08:24:29 +0000

Hello, i have silly question. How intuitively explain that cos x/x is undefind?
There is graf> http://www.wolframalpha.com/input/?i=Plot%5B{cos%5Bx%5D%2C+x}%2C+{x%2C+-1.0%2C+1.0}%5D

thx

By: Kalid

Kalid — Thu, 09 Jun 2011 00:29:53 +0000

@Dave: Great question. I can’t say I’m completely comfortable with limits, but I think you can jump back and forth (the Keisler Calculus book has some examples like this I believe). I think the bigger goal is to figure out what is being said, i.e. “What does this equation equal, within some level of tolerance?”. Limits and infinitesimals are two ways to define that tolerance threshold, but infinitesimals are “easier” in that it’s built in (and you don’t need to explicitly define epsilon, delta, etc.).

By: Dave

Dave — Wed, 08 Jun 2011 14:44:18 +0000

Hey, Kalid, I’ve just got a quick question to ask.

If you learn calculus via the use of infinitesimals, is it possible to then make the leap over to using limits? While I doubt it would happen, I’d like to be an amateur mathematician in the vein of Fermat some time and develop proofs (more as a beauty thing, to be honest), but writing in a fashion that is contrary to the norm is rather like handing out Spanish pamphlets in an English neighborhood- they might understand, but they won’t like it.

So, yeah, can you jump from infinitesimals over to limits? From what I can tell, limits are mainly used because they’re easily to rigorously define an to keep the constructivist camp from yelling at you.