Another Pointless Discsussion! :-)
On to a tangential question...
It is all relative to the scale that is being examined. But points are dimensionless. So I take issue with the precise idea that a tangent can be determined by two points that are in close proximity. "Leibniz defined it as the line through a pair of infinitely close points on the curve." Wikipedia. The problem is, no matter how close two points are, there are an infinite number of points between two inifnitely close points. A better definition is to think in terms of tuples. A curve consists of an infinite set of tuples. For each tuple in that curved line, there only one tuple at a time can be concurrent with a tuple in an intersecting straight line. When that straight line intersects precisely that one point tuple in the curved line, then the straight line is a tangent. Now, practically (why differentian and integration was invented), there had to be a way to call "Close Enough" and converge to an answer. That works when the measurements needed to find that precise tuple, are below the level of accuracy possible for the scale you are working with.
So it matters that points are dimensionless and infinte in all mathematical applications. But to be practical, and make sense of a problem, points have to be defined in a way that is useful to the scale that the problem exists in. For example, if you zoom into a circle far enough, geometrically, the segment you are examining will appear to be a straight line, because the remaining curve falls below measurability. Practically, it becomes a straight line. Mathematically, the infinite set of tuples representing the curved line, is still curved.