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

For \(f: \mathbb{R}\rightarrow \mathbb{R} \) and \( g:\mathbb{R}\rightarrow \mathbb{R}\)

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

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