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!