Question

Solve: $\underset{x\to 0}{lim}{(1+lo{g}^2\cos \frac{x}{2}\cos x)}^2$.

Answer 1

$\underset{x\to 0}{lim}{(1+lo{g}^2\cos \frac{x}{2}\cos x)}^2$

$=\underset{x\to 0}{lim}{(1+{\left(log\cos \frac{x}{2}\cos x\right)}^2)}^2$

$=\underset{x\to 0}{lim}{(1+{(log\cos \frac{x}{2}+\log \cos x)}^2)}^2$

$=1+\underset{x\to 0}{lim}{(log\cos \frac{x}{2}+log\cos x)}^4+2$ $\underset{x\to 0}{lim}(log\left(\cos \frac{x}{2}\right)+log\cos x)$

$=1+0+0$

$=1$

$\therefore \underset{x\to 0}{lim}{(1+lo{g}^2\cos \frac{x}{2}+\cos x)}^2=1$.