symptotic Curves with No Points of InflectionWhere exactly does a mountain meet a plane? The following is my definition of a unique point for all curves which approach the x and y axis asymptotically.
Let the radius of curvature (blue) proceed to a unique point (x,y) such that a line (red= tangent to the curve) drawn from the end point of the 'inverse" radius of curvature, meets the curve at (x=0, y=infinity).
I define this point as the beginning of the mountain and end of the plane based on experience with the "learning curve".
Suppose you wish to learn to play golf as well as Nicklaus, Palmer, etc. You have no idea initially what you must do to play at this level. When should you give up given that you actually aren't that serious (you just think you are)?
The best answer, based on my experience is to practice until you can "see" the monseiurs farther up the learning curve. And you can only see them when you have proceeded to near the point described, i.e. they are reasonably close to perfection as to be functionally indiscernable from that state.
For human lifespans, the amount of time necessary to reach this point is remarkably consistent ... about 2000 hours of genuine practice. This is the general amount of time required to "get the hang" of just about any endeavor (not to become expert but rather to know the subject well enough for reasonably accurate judgements).
5000 hours will get you into the tournament (if you are diligent).
And 10,000 hours will allow you to compete at the "Monseiur" level. This amount of time generally weeds out the indifferent, complacent and incompetent.
They just can't hack it.