Limit , will you use L'H rule ?

$\displaystyle{f\left( x \right) =\cos ^{ -1 }{ \left( \cos { x } \right) } \\ g\left( x \right)=\sqrt { \left\{ x \right\} -{ \left\{ x \right\} }^{ 2 } } \\ }$

For $f: \mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$

Find $\displaystyle{\lim _{ x\rightarrow \infty }{ \frac { \displaystyle \int _{ 0 }^{ x }{ f\left( t \right) \ \mathrm dt } }{ \displaystyle \int _{ 0 }^{ x }{ g\left( t \right) \ \mathrm dt } } } }$