Is this a correct way to see differentials in calculus? I'm trying to get a grasp on the differentials concept and I arrived to this conclusion, but I'm not sure it's correct since I haven't found exactly this way of defining them anywhere.
I'm pasting a picture of my notes because I'm not sure how to express this otherwise.

From my understanding, defining $dx$ and $dy$ this way gives me the following properties:


*

*Differentials are limits, not numbers, so they must be dealt with using algebra of limits

*Differentials represent infinitesimal quantities


I want to check if these deductions are correct, misguided or maybe plain wrong.

I don't see any definitions of $dx$ and $dy$ there.  There are ways of defining differentials $dx$ and $dy$, but it's not as limits of $\Delta x$ and $\Delta y$.  You might look at this question
 and its answers.
