Technically, f(x) converges to y as x -> infinity if for every epsilon there is an X such that |f(x) - y| < epsilon for all x > X ("the error becomes arbitraril

Technically, f(x) converges to y as x -> infinity if for every epsilon there is an X such that |f(x) - y| < epsilon for all x > X ("the error becomes arbitrarily small"). For instance, 1/x converges to 0 as x -> infinity (pick X = 1/epsilon), but the sequence 0, 1, 0, 1, 0, ... does not converge to any y (pick epsilon < 1/2 to see this). In particular, the sequence r, r^r, r^r^r, ..., where r^2 = 2, does converge (to 2; the first terms are approximately 1.41, 1.63, 1.76, ...); the sequence 2, 2^2, 2^2^2, ..., however, does not (the first terms are 2, 4, 16, ... - and it only grows faster from there.)