The value of limn→∞ cos⁡x2cos⁡x4cos⁡x8…cos⁡x2n, is

We know that cos⁡Acos⁡2Acos⁡4A…cos⁡2n−1A=sin⁡2nA2nsin⁡ATaking A=x2n, we get cos⁡x2ncos⁡x2n−1…cos⁡x4cos⁡x2=sin⁡x2nsin⁡x2n∴ limn→∞ cos⁡x2cos⁡x4⋯cos⁡x2n−1cos⁡x2n=limn→∞ sin⁡x2nsin⁡x2n=limn→∞ sin⁡xxx/2nsin⁡x/2n=sin⁡xx