Update: H/t David K for pointing out that my assumption that I can force $a^2+b^2=r^2$ is wrong. This led to analyzing a cubic equation which is now moot, but I think the bulk of the question remains sensible.
I think I'm making some mistakes in the below, but I'm not sure where. Even if there are mistakes, I'm thinking there might not be any practical point to it, but I'm curious.
I think it's possible to find the slope of a tangent line without using limits in some cases. Further it's possible to generalize the procedure to many common curves. I am having trouble with a proof of this and I'm not sure where I am going wrong.
Proof we can find the slope of a tangent line to a parabola without taking a limit:
1) First we consider a circle. If we know the center of a circle and we know a point on that circle, we know the equation of a line connecting the center to that point. The slope of the line perpendicular to this line is the slope of the tangent line to the circle at that point. So we have already proven the thesis in the case of a circle.
2) A circle is tangent to some other curve if it only intersects that curve at a single point.
3) A circle can be constructed to be tangent to certain points on certain curves. Tangent circles share tangent lines with the curves to which they are tangent. So the procedure above can find tangents to those curves by constructing the correct tangent circle.
Parabola case: Consider $y=x^2$.
We want a circle such that $(x-a)^2+(x^2-b)^2=r^2$
We either have 2 complex roots or no real solutions. Assuming we have any real solutions, having a unique one requires that we have a repeated root.
So to find the tangent to the curve at $x_0$ we want to divide the quartic by $(x-x_0)^2$. We'd want to have the remainder terms be zero and the discriminant of the resulting quotient to be negative.