Find the limit, as N goes to infinity, of the broken part (i.e., x-integer(x)) of: (2+Ö2)^N
 

The limit is 1:

(2+Ö2)^N + (2-Ö2)^N is an integer number (check binomial expansion).
(2-Ö2)^N converges to 0 since 2-Ö2 is less than 1.