limn→∞ limm→∞ cos2m(n!pix)x belongs to R
prove the above function =1 if x is rational
function =0if x is irrational
-
UP 0 DOWN 0 0 2
2 Answers
1Some facts : -
1 >
" n ! " contains all the integers ranging from " 0 " to " n " . So , as " n " tends to infinity , all the integers you can ever think of are included .
2 >
If we let " x = ab " , i . e , a rational number , then naturally , " b " is included in " n ! " , hence , " n ! . x " becomes an integer .
The solution : -
Consequently , the argument of " cos " becomes a multiple of " π " in the first case . Now , since the power of the " cos " is always even , hence , the limit eventually becomes " [ cos 2 ( I π ) ] ∞ = 1∞ = 1 " , where " I " is any integer .
Similarly , if " x " is an irrational number , then the argument inside the " cos " does not equal a multiple of " π " . Now , " cos " being always less than or equal to 1 , is a fraction valued function . Now , it is well known that a fraction whose power tends to infinity tends to zero . So , in this case , the given function equals " 0 " .
