finding limit with $\cos$ function occur $n$ times Finding $\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (1-\cos x))))}{x^{2^n}}$
where number of $\cos$ is $n$ times
when $x\rightarrow 0$ then $\displaystyle 1-\cos x = 2\sin^2 \frac{x}{2} \rightarrow 2\frac{x}{2} = x$
so $1-\cos (1-\cos x) = 1-\cos x$
some help me., thanks

$$\lim_{x\rightarrow 0}\frac{1-\cos x}{x^2}=\frac12$$
thus $1-\cos x\to\dfrac12x^2$ with substantiation 
$$\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (1-\cos x))))}{x^{2^n}}$$
$$=\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots (\dfrac12x^2))))}{x^{2^n}}$$
$$=\displaystyle \lim_{x\rightarrow 0}\frac{1-\cos(1-\cos(1-\cos(1-\cdots \cdots \dfrac12(\dfrac12x^2)^2)))}{x^{2^n}}$$
$$=\displaystyle \lim_{x\rightarrow 0}\frac{\left(\dfrac12\right)^{2^n-1}x^{2^n}}{x^{2^n}}$$
$$=\left(\dfrac12\right)^{2^n-1}$$
