<s>[INST] Is this true: $\lim_{x\to 0} \frac{1-(\cos x)^n}{x^2}=\frac n2$ Does this statement hold $\forall n\in \Bbb R$?
$$\lim_{x\to 0} \frac{1-(\cos x)^n}{x^2}=\frac n2$$
I know that it holds for $\forall n \in \Bbb N$, because $1-(\cos x)^n=(1-\cos x)(1+ \cos x + ... + (\cos x)^{n-1})$, and $\lim_{x\to 0} (1+ \cos x + ... + (\cos x)^{n-1})=(1+1+...+1)=n$, but does that stand for, for example, $n=\sqrt2$?
If it does, how do I prove it?
 [/INST] Note that for $n\in \mathbb{N}$
$$1-(\cos x)^n=(1-\cos x)\stackrel{\color{red}{\text{n terms}}}{(\cos^{n-1}x+\cos^{n-2}x+...+1)}$$
thus since $\cos x\to 1$
$$\frac{1-(\cos x)^n}{x^2}=\frac{(1-\cos x)}{x^2}\stackrel{\color{red}{\text{sum up to n}}}{(\cos^{n-1}x+\cos^{n-2}x+...+1)}\to\frac12\cdot n=\frac n2$$
For $n=a\in \mathbb{R}$ note that
$$\cos x=1-\frac{x^2}{2}+o(x^2) \quad (1+x)^a=1+ax+o(x) \\\implies \left(1-\frac{x^2}{2}+o(x^2)\right)^a=1-a\frac{x^2}{2}+o(x^2)$$
then
$$\frac{1-(\cos x)^n}{x^2}=\frac{1-1+a\frac{x^2}{2}+o(x^2)}{x^2}=\frac{a}2+o(1)\to\frac{a}2$$
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