# Math Help - limit

1. ## limit

x belongs to [0,1]

limit n->infinity (limit m-> infinity cos 2n (m! pi x) )=?

its value is 0 if x is irrational
and it takes 1 if x is rational.

$\lim_{m\to\infty}\left(\lim_{n\to\infty}\left(\cos (m!\pi x)^{2n}\right)\right)$.
If x is rational, then m!x is eventually an (even) integer. If x is irrational, then $\cos(m!\pi x)\in(-1,1)$, so the inner limit is 0.