This is in continuation to my earlier post at : Tangent to Cubic curve has positive slope.
It is discussed in earlier post (& before that) that given a cubic curve $y = x^3$, if tangents are drawn from a point on the curve to the cubic curve, then there is an infinite sequence of such tangents' $x$-coordinates with G.P. for $x$-coordinate ratio being $=-2$, i.e there is an infinite sequence of tangents with their $x$ coordinates having a geometric ratio of $-2$.
The angle of $90^\circ$ (for tangents made from 3rd quad to first), or $270^\circ$ (for vice-versa) is approached very quickly for a small starting $x$ value, say $x=+0.11$; it takes 3-4 steps in the sequence to get slope of a vertical line (i.e. $90^\circ$, or $270^\circ$).
Also, it is shown in code file of the linked post at : http://py3.codeskulptor.org/#user301_QKlzKmDtjSmgRMn.py, that the angle is $270^\circ$ for large $x$-coordinate ($x=7.04$ in the $n=6$ step) point's tangent's slope.
I am at pain to imagine how angle of tangent from 1st to 3rd quad. for large positive $x$ (say, $x=7.04$) to large $-x$ value (now, $-2\cdot x=-14.08$) is $270^\circ$.
It is confusing to think that such connecting line is a vertical line.
Request some tool that enables one to visualize such un-intuitive angle. If there is a site, applet, tool, or logic to make one see the same intuitively; then please tell.