Classical Limits vs. Non-Standard Limits
One of the most important and fundamental concepts taught in modern Calculus classes is that of the Limit. I have discussed this idea once before, but I thought I would revisit it, here. In that first article, I noted that the classical definition for a Limit is fairly complex and that we can utilize a more intuitive notion of infinitesimals to accomplish the same task, insofar as derivatives are concerned. However, there are other uses and purposes for limits, in mathematics, so we would not want to simply omit them entirely, even using a non-standard approach to Calculus lessons.
Thankfully, even the very difficult and complex definition of “limit” can be simplified and made easier to understand by use of non-standard infinitesimals.
. We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, 
, there are an infinite number of Rational numbers which are greater than 
, which is not equal to any Rational number. There are Rational numbers which are greater than 
Boxing Pythagoras 