Visualizing infinitesimals

Question

I am having problems visualizing infinitesimals. Like in a curve, when we integrate we can adjust in it a perfect infinitely thin rectangle. But how do I visualize it? Every time I try to visualize it, I always notice some gap area or error in the curve. Also in differentiation, $ dx $ is supposed to an infinitely close step. But every time I draw a line, I can cut it into smaller line and that becomes $ dx $. So I am facing bit problema visualizing infinitesimals.

Answer 1

sometimes there is error/gap, however as the dx (infinitesimal) tends to 0 this error tends to 0 faster (quadratically, cubicly...). Yes you may be able to make dx smaller which is what i think you are saying by "cut it into smaller line" and thats correct because thats the whole idea of a Reimann Sum which is what approximates an integral as the dx basically becomes 0 (almost)

Answer 2

There's an easy workaround: forget infinitesimals, learn the definitions of limits, derivatives, integrals,... and work with them. I'll admit there may be some collateral damage: after several years of exercise and experience, you may start seeing those "infinitesimals", and which "gaps" you can safely ignore.

Alternative: you may study nonstandard analysis, making infinitesimals a rigorous notion. But there are several systems (Robinson, Nelson,...), none of them being trivial, all of them having their own pitfalls.

Answer 3

Just like $x$ has no fixed value (it's variabel), so is $dx$. So you shouldn't be irritated by the fact that, even if you manage to visualise $dx$, there is still a "smaller $dx$". The way I visualise $dx$ is by contrasting it to $\Delta x$ with a looking glass that zooms in "infinitely deep". Here is a sketch: