In Rational Points on Elliptic Curves (Silverman, Tate), they introduce projective curves and then state:
The intuition to keep in mind: an affine curve is somehow "missing" some points that lie out at infinity and the points that are missing are the limiting directions as one moves along the curve toward infinity. (pg. 272)
I was wondering, doesn't this mean that all the curves have to converge as they approach infinity? Because we can graph function in $R^2$ that don't converge (e.g. trig functions), so how could there be a "limiting tangent direction"?