Chalte Chalte weird limit

these were done some time back in forum..u can find them..

here is the link for 1st two
http://www.targetiit.com/iit-jee-forum/posts/area-262.html

ok got the 1st....wht abt 2nd and third!!!

@manipal can u give the full solution!!!!

1)f(x)=lim_n->∞cos(n! 2Πx)
2)g(x)=lim_n->∞cosⁿ( 2Î x)
3)h(x)=lim_n->∞{cos(nx) + 2}^1/n

(1)
CASE I: x is rational
Let x=p/q where p,q is an integer
as n→∞, so n! will cancel out any possible value of q
So the expression inside cos will be cos(2mÎ ) where m is an integer
So, its value will be 1

CASE II: x is irrational
Then for no value of n! the irrational value of x can be canceled. So,
here cos() can take any value from -1 to 1 depending upon the value of x

So, domain of x = R
and range = (-1,1]

(2)

x can take any value. And the range will be [-1,1]

(3) x can take any value and range will be 1