Thread: Convergence of continuous function

My professor hinted that this problem would be on the final, so any help would be greatly appreciated.

Find:
limit as n goes to infinity of (limit as k goes to infinity of cos((n!*pi*x)^2k))

I know since it's continuous, I should be able to move the limits inward, but I don't know if that's the right way to go about this problem. I'll take any suggestions please.

First we 'fix' n and define...

(1)

... and leaving undefined for all other values of x. Now what is the limit when n tends to infinity of ?...

Kind regards