I am new to Asymptotic analysis so please bear with me and i apologize if the following question is not well formed or is trivial. I am trying to figure out Asymptotic behavior of the following two functions. $\phi_1(x) = 1 - \cos\left(1-\cos\left(x\right)\right)$ and $\phi_2(x) =x\sin\left(\sin x\right) -\sin^2 x$ when x is small.
Now i am aware of the following two limits $\lim _{x\rightarrow 0}\dfrac {\sin x} {x}=1$ and $\lim _{x\rightarrow 0}\dfrac {1-\cos x} {x}=0$
I suspect that $\phi_1(x)$ and $\phi_2(x)$ are of the fourth and the sixth order respectively, but i am unsure how to use the insight from the limit expressions to show this. Any help would be much appreciated.