1.4 Infinitely small quantity [limit]
1.4.1 Introduction
When early mathematicians learned about infinitesimals, they understood infinitesimals as "numbers smaller than everything but greater than 0" and "numbers as small as they are." Today's beginners usually understand infinitesimals this way before formally learning infinitesimals. In most questions, there is no problem with this understanding, so now comes the problem.
The first question: Suppose△x is infinitesimal, then is 2△x also infinitesimal? If 2△x is also infinitesimal, and 2△x is both greater than 0 and △x, it is the same as "less than everything" But the understanding of "numbers greater than 0" is contradictory. So infinite small cannot simply be understood as "a number smaller than everything but greater than 0".
The second question: They are all numbers as small as they need to be, so is the size comparison between them meaningful?
Give an example to illustrate. What is the cube of 1 or 1? What is the cube of 1.001? Definitely a little bigger than 1, but certainly not much bigger. Use a calculator to calculate it: 1.003003001. His decimal part can be decomposed into three parts
0.003003001 = 0.003 +