When differentiating y w.r.t. x when y is a function of x, we write:
$\lim\limits_{\delta x \to 0}(\frac{\delta y}{\delta x})=\frac{dy}{dx}$
Instead, can we write the following (since as $\delta x$ approaches 0, so does $\delta y$)?
$\frac{dy}{dx}=\lim\limits_{\delta y \to 0}(\frac{\delta y}{\delta x})$
If this is incorrect, why?
The answer is no.
Let $$y=x^2$$
We have $$\delta y = ( x+\delta x )^2 - x^2 = \delta x ( 2x + \delta x)$$ As you notice, $\delta y =0$, if $2x + \delta x =0$ without $\delta x$ approaching $0$