Precisely how is "infinitesimal" calculus meaningfully different from "limit-based" calculus? How exactly is "infinitesimal" calculus different from "limit-based" calculus? I've heard people argue over which is the "best approach" to the subject, and I've read numerous books and articles that emphasize the distinction, yet I've never seen someone lay out precisely what makes the approaches unique.
How would a class in "infinitesimal" calculus differ from a typical calculus class?
Skimming a few books, articles, and Wikipedia, it doesn't seem to me that they're different approaches at all.
 A: Honestly, I'm probably not the most qualified to talk about this, but I'll give it a shot.
Infinitesimal calculus is really how it was thought of when it was first created. We know the area under a curve because we just add up all of these infinitesimally small slices. However, and really think about this, how small is an infinitesimal?
This is why I said that this is how calculus was thought of when it was first created. We don't really think of it like this anymore, although it is certainly taught like this sometimes. Personally, I still quite enjoy this idea of an infinitesimal. It's easy to comprehend this way, but that doesn't mean it's the most mathematically sound, because there is no way to quantify how small this infinitesimal is.
This is called the Archimedean Property. You can look it up, but it really just boils down to this: no matter how small x gets, there will always be a smaller x. So, since infinitesimals aren't good enough, calculus had to be entirely redefined soon after its invention.
This brings us to limits. Limits are nice, happy, mathematically sound ways to write our stuff. It's not that infinitesimals are necessarily wrong, but there's better ways to write it when it comes to concrete, mathematical reasoning. The idea is still the same.
I acknowledge that this post is prone to errors, so if you see anything, let me know and I'll gladly correct it. I'm sure it could be more in-depth, but I believe that the general idea is there.
Let me know if this clears it up.
A: This may be a trivial disagreement, over whether two approaches that are visibly somewhat different are significantly different. But for what it's worth, the infinitesimal approach says that the derivative $f'(x)$, for instance, is the standard real number to which $\frac{f(x+h)-f(x)}{h}$ is infinitely close whenever $h$ is infinitely small. This is visibly not the same as the limit-based approach, which says that $f'(x)$ is the number so that, no matter what $\varepsilon>0$ is chosen, there is a $\delta>0$ so that $f'(x)$ is within $\varepsilon$ of $\frac{f(x+h)-f(x)}{h}$ whenever $h$ is within $\delta$ of $0$. Similarly, in the infinitesimal approach, $\int_a^b f(x)dx$ is the standard real number to which the Riemann sum of $f$ over any infinite number of infinitely narrow rectangles is infinitely close, which is not the same as the standard definition.
