# Convergence of continuous function

• May 2nd 2011, 08:21 PM
abisu
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.
• May 2nd 2011, 08:45 PM
chisigma
First we 'fix' n and define...

http://quicklatex.com/cache3/ql_238a...ce29c96_l3.png (1)

... and leaving http://quicklatex.com/cache3/ql_1637...5b9b4a4_l3.png undefined for all other values of x. Now what is the limit when n tends to infinity of http://quicklatex.com/cache3/ql_1637...5b9b4a4_l3.png?...

Kind regards

$\chi$ $\sigma$