Use the taylor series approximation. i.e. for small t,
1 - cos(t^2) ≈ 1 - (1 - (t^2)^2 / 2!) = t^4 / 2
Then integrate that from 0 to x, so you get
t^5 / 10 + C
from t=0 to t=x, which gets you
x^5 / 10
Then multiply by the 1/x^5, which gives 1/10.
(But you have to prove that the estimate is allowed.)