Could one say that a line L is the tangent to curve C at point P just in case :
(1) L is not identical to C [ unnecesssary condition according to the answer given below]
(2) L is the straight line an object would follow in case this object would leave C at P , continuing freely its movement in virtue of inertia
(3) this object would follow the same line L either in case it would come from the left or in case it would come from the right ( in the cartesian plane).
Would this intuitive explanation of " tangent line to a curve" be in agreement with the rigorous definition of the tangent concept?
Note : condition (2) would prevent the X-axis from being the tangent at (0,0) to the curve f(x) = |x| ; on account of conditon (3), it is f(x) = -x as well as f(x) = x that would not qualify.
I would say yes, since from a physical point of view you are basically describing the instantaneous velocity. Instantaneous velocity is always tangent to the trajectory of an object:
About point 1: The tangent line of a point on a differentiable function is the line that goes through that point and has the same slope as that point. Thus, the tangent of a straight line is the straight line itself.