This is more like a comment to the question in Mathematics that starts with:
"If you have a line, and it goes on forever, and you choose a random point on that line, is that point the center of that line? And if you ..."
The answer provided by the panelist, as well as the initial question, assume that one can distinguish between points at infinity. As far as Math goes however, one cannot do that, and this is the reason the limit for cos(phi) does not exist, as phi goes to infinity. Revisiting the argumentation provided by the panelist, the error starts with the 'definition' of the distance between a fixed point and infinity - this distance cannot be defined, and therefore it cannot be compared (at least, as math goes).
A somewhat similar problem can be stated, without the pitfalls of the infinity concept, for a point on a circle, or any closed curve.