I am reading about infinitesimals and came up to the following calculation: (Suppose f is differentiable at point x = a)

\(\displaystyle Δf - dy = Δf -f'(a)Δx = (\frac{Δf}{Δx} - f'(a))\cdot Δx = ε \cdot Δx \)

Now as Δx approaches 0 the fraction \(\displaystyle \frac{Δf}{Δx}\) approaches f'(a) and therefore the portion inside the parentheses approaches 0. (Based on the textbook)

But the definition of the fraction \(\displaystyle \frac{Δf}{Δx}\) as Δx approaches 0 is exactly f'(a).

So why is it that ε approaches 0 instead of ε equals to 0.

Thanks.