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I was introduced to Calculus by the online series on it by Grant Sanderson (3Blue1Brown's owner) called Essence of Calculus.

In his videos, he treats $dx$ as $\Delta x$ that approaches $0$ and $dy$ as the corresponding change in $y$ ( i.e. $\Delta y$). He especially mentions in one of his videos that he doesn't like to treat $dx$ and $dy$ as infinitely small quantities but rather finite quantities that approach $0$, which is similar to the idea behind limits rather than infinitesimals. In that very video, he defines $\dfrac{df}{dx}$ as what the slope of the line joining $(x,f(x))$ and $(x+\Delta x, f(x+\Delta x))$ approaches as $\Delta x \rightarrow 0$, which is another way of saying that : $$\dfrac{d}{dx}f(x) = \lim_{\Delta x \rightarrow 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}$$ On the other hand, he treats $\dfrac{dy}{dx}$ as a ratio between $dy$ and $dx$ which feels more similar to the infinitesimals approach. He also derives the fundamental theorem of Calculus, namely : $$\int_a^bf(x)dx = \int_0^bf(x)dx - \int_0^af(x)dx = F(b)-F(a) \text{, where : } F'(x) = f(x)$$ using geometric intuition which feels more like an infinitesimal-related approach.

Overall, I feel that his approach to Calculus is a combination of the limits and infinitesimal approach but is more inclined towards limits rather than infinitesimals and while some of my peers agree with me, many don't. I would like to know what Math SE users think of the same.

Thanks!

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    $\begingroup$ It would be best to add a link to the video here $\endgroup$ – BLM Aug 15 at 15:01
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    $\begingroup$ I think the modern approach is generally to describe everything in terms of limits. $\endgroup$ – Michael Morrow Aug 15 at 15:14
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I'll flesh out @MichaelMorrow's comment, with a qualification: the standard modern approach is to describe everything in terms of limits.

Historically, calculus grew out of a desire to understand what empirically seem to be continuous processes, but at a time when we hadn't fully developed the theory of limits. This is unfortunate, because in modern language $f^\prime(x)$ is literally defined as $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. In other words, $f^\prime(a)=L$ is shorthand for $\forall\epsilon>0\exists\delta>0\forall h\left(0<|h|<\delta\to\left|\frac{f(a+h)-f(a)}{h}-L\right|<\epsilon\right)$, a statement that requires no infinitely large or small quantities at all. Before we could put it like that, it was somewhat vaguer (at least according to critics of the time), with the idea of secant lines' gradients approaching a tangent line's gradient.

But we are not slaves to history. The standard modern approach is to define limits, and then define differentiation and define integration, both in terms of limits. The nonstandard modern alternative is to introduce some axioms for "infinitesimal" quantities which, while not among the real numbers, give the same results as the above treatment in terms of limits of real-valued functions. In particular, it allows us to say $df(x)=f^\prime(x)dx$ rather than having to say $\int_{x=a}^{x=b}df(x)=\int_{x=a}^{x=b}f^\prime(x)dx$. It also requires "infinitesimals" to anticommute, as discussed here. That we can take such an approach is interesting, but it's not how we normally do things.

What 3blue1brown does isn't quite the same as anything I've discussed so far. I don't merely mean "oh, he uses different definitions/axioms". He's not merely giving definitions and proofs; rather, he tries to motivate specific ways of thinking about how you'd define and prove things, when the fog is unclear. (That's true across his excellent YouTube channel.) This is an important part of mathematics too; in fact, it's roughly how we discover how we "should" define & prove things. If you want to learn a topic in mathematics, there is no substitute for learning what came out on the other side of the fog, but it's instructive to look at both sides.

Edit: as @pash has noted, when I refer to certain approachs as nonstandard, I don't mean that adjective as a technical term, merely that "these approaches exist, but they're definitely not what we normally do" (I was leaning on Morrow's use of "standard"). Like anything else I discussed, NSA in general only makes sense in terms of how it uses limits. So the take-home message is you must learn limit-based definitions to get anywhere in calculus.

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  • $\begingroup$ I have to say, this answer is absolutely perfect. When I think about differentiation in terms of infinitesimals, I get the idea that we are approximating the slope of the tangent but thinking in terms of limits makes it feel exact, for some reason and not approximate. Am I thinking of it right? $\endgroup$ – Rajdeep Sindhu Aug 15 at 19:35
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    $\begingroup$ @RajdeepSindhu Let's take an example. The secant on $y=x^2$ from $x=a$ to $x=a+h$ has gradient $2a+h$, which has $h\to0$ limit $2a$. Furthermore, the value of $y$ on such a secant at $x=b$ is $a^2+(2a+h)(b-a)$, which as $h\to0$ approaches $a^2+2a(b-a)$, which is the $y$-coordinate at $x=b$ of the gradient-$2a$ line through $(a,\,a^2)$. $\endgroup$ – J.G. Aug 15 at 20:01
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    $\begingroup$ Nonstandard approaches are not typically nilpotent, i.e., it is not typically true in nonstandard analysis that $dx^2 = 0$. The only formal nilponent theories are necessarily nonconstructive (see e.g., John Bell’s Primer of Infinitesimal Analysis) and are quite different from the formulations of NSA following Robinson or Nelson. $\endgroup$ – pash Aug 16 at 6:23
  • $\begingroup$ Informal methods (à la the physicists) are unhelpful here, IMO, because infinitesimals-vs-limits, and analysis itself, is fundamentally about formalism. I think a good answer would show how NSA defines all of the relevant concepts without resorting to the sidelong approach of logically quantified epsilons and deltas. Indeed “infinitesimals versus limits” is a naive characterization: limits are as fundamental a concept in NSA as in ordinary analysis. But the logical apparatus differs, to the effect that NSA gives you a definition of “sufficiently small” that cleans up the usual definitions. $\endgroup$ – pash Aug 16 at 7:02

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