Question

limx tends to 0 1-cos (1-cosx)/x^4

Answer 1

You could try and use the fact that

limx→01−cosxx2=12limx→01−cos⁡xx2=12

This can be proved easy, using l'Hospital, or just writing 1−cosx=2sin2x21−cos⁡x=2sin2⁡x2.

So returning to your problem you can write your limit as

limx→01−cos(1−cosx)(1−cosx)2⋅(1−cosx)2x4limx→01−cos⁡(1−cos⁡x)(1−cos⁡x)2⋅(1−cos⁡x)2x4

and use two times the limit described at the beginning of the answer.

l'Hospital also works but you'd probably have to differentiate four times until you get the result.