What is Asymptotic Point in overlaod protection?

Not 100% sure what you are asking or why it would be important.

If I remember my analysis courses correctly, that would be the point in the I^2t (Asymptotic) curve at which the increase becomes more curved than linear, is that right? If so, I've always felt that it is sort of a judgment call (unless you are a math whiz and plan to spend weeks analyzing the statistical relevance of the curve shape).

But my next point is, why is this important? It's a mathematical issue but means nothing as far as the motor is concerned because the only critical issue is whether the OL trip curve remains under the damage curve of the motor at any given point (usually the rotor damage curve is lower than the stator damage curve). We use I^2t as the OL curve only because it was found statistically to be very close to what happens thermally in the motor, however it is not exact.

I'm just guessing here, but if you were trying to figure out where you might "play around safely" by avoiding that Asymptotic point on the mathematic model, you are almost literally playing with fire. A more relevant analysis would be to have the lowest damage curve and plot it against the OL curve.