<s>[INST] 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
 [/INST] Take the simple case 
$$\frac{1-\cos(1-\cos x)}{x^4}$$
$$=\frac{1-\cos(1-\cos x)}{(1-\cos x)^2}\left(\frac{1-\cos x}{x^2}\right)^2$$
$$\rightarrow \frac{1}{2}\left(\frac{1}{2}\right)^2$$
Can you see how to do the induction ?
So you have in general if $f(n)=1-\cos(1-\cos(1-\cos(1-\cdots \cdots (1-\cos x))))$
$$\frac{f(n)}{x^{2^n}}=\frac{f(n)}{f(n-1)^2}\left(\frac{f(n-1)}{x^{2^{n-1}}}\right)^2\rightarrow \frac{1}{2}L_{n-1}^2$$
And if we write $L_n=\left(\frac{1}{2}\right)^{e_n}$
we get the recurrence relation 
$$e_n=2e_{n-1}+1$$ which means 
$$e_n=2^n-1$$
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