Question

The value of $\underset{x\to 0}{lim}\frac{32}{x^8}(1-\cos \frac{x^2}{2}-\cos \frac{x^2}{4}+cos\frac{x^2}{2}\cos \frac{x^2}{4})$

Answer 1

$\underset{x\to 0}{lim}\frac{32}{x^8}(1-cos\frac{x^2}{2}-cos\frac{x^2}{4}+cos\frac{x^2}{2}cos\frac{x^2}{4})$

$=\underset{x\to 0}{lim}\frac{32}{x^8}\left[\right(1-cos\frac{x^2}{2})-(cos\frac{x^2}{4}-cos\frac{x^2}{2}cos\frac{x^2}{4}\left)\right]$

$=\underset{x\to 0}{lim}\frac{32}{x^8}\left[\right(1-cos\frac{x^2}{2})-cos\frac{x^2}{4}(1-cos\frac{x^2}{2}\left)\right]$

$=\underset{x\to 0}{lim}\frac{32}{x^8}\left[\right(1-cos\frac{x^2}{2}\left)\right(1-cos\frac{x^2}{4}\left)\right]$

$=\frac{32}{x^8}\left[2\right(\frac{x^2}{4}{)}^2\left]\right[2\left(\frac{x^2}{8}{)}^2\right]$

$=\frac{32}{x^8}\left(\frac{x^8}{256}\right)$

$=\frac{1}{8}$