Question

if `f(n,theta)=prod_(r=1)^(n) (1-tan^(2)(theta)/(2^(r)))` then the value of `lim_(thetato0) [lim_(ntoinfty) f(n,theta)]`, (where `[x]` represent the greatest integer function) is

Answer 1

`prod_(r=1)^(n)(1-tan^(2) (theta)/(2^(2)))`
`=(2tan``(theta)/(2))/(tantheta).(2tan``(theta)/(2^(2)))/(tan``(theta)/(2))(2tan``(theta)/(2^(3)))/(tan``(theta)/(2)).....(2tan``(theta)/(2^(n)))/(tan``(theta)/(2^(n-1)))`
`f(n,theta)=(2^(n)tan``(theta)/(2^(n)))/(tantheta)=(theta)/(tantheta)` as `ntoinfty`
`underset(thetato0)(lim)[(theta)/(tantheta)]=0` as `tanthetagttheta` as `thetato0`