Question:
If $\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$
then the value of $\mathrm{k}$ is_______________
Solution:
$\lim _{x \rightarrow 0}\left\{\frac{1}{x^{8}}\left(1-\cos \frac{x^{2}}{2}-\cos \frac{x^{2}}{4}+\cos \frac{x^{2}}{2} \cos \frac{x^{2}}{4}\right)\right\}=2^{-k}$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\left(1-\cos \frac{x^{2}}{2}\right)}{4\left(\frac{x^{2}}{2}\right)^{2}} \frac{\left(1-\cos \frac{x^{2}}{4}\right)}{16\left(\frac{x^{2}}{4}\right)^{2}}=\frac{1}{8} \times \frac{1}{32}=2^{-k}$
$\Rightarrow 2^{-8}=2^{-\mathrm{k}} \Rightarrow \mathrm{k}=8$