Limits and Deri...

$$\displaystyle \lim _{ x\rightarrow \dfrac { \pi  }{ 2 }  }{ \dfrac { \sin { x }  }{ \cos ^{ -1 }{ \left[ \dfrac { 1 }{ 4 } \left( 3\sin { x } -\sin { 3x }  \right)  \right]  }  }  } $$, where [.] denotes greatest integer function is :

$$Im _{  }{ \left( \dfrac { 1 }{ 1-\cos { \theta  } +i\sin { \theta  }  }  \right)  } $$ is equal to

$$\lim _ { x \rightarrow 0 } \frac { \sqrt [ 3 ] { 1 + \sin x } - \sqrt [ 3 ] { 1 - \sin x } } { x } =$$

$$\displaystyle\underset{x\rightarrow \dfrac{\pi}{4}}{Lt}\dfrac{\sqrt{2}-\cos x-\sin x}{(4x-\pi)^2}=?$$

$$\underset { \theta \longrightarrow 0 }{ Lt } \dfrac { 3tan\theta -tan3\theta  }{ { 2\theta  }^{ 3 } } =$$

$$\lim _ { x \rightarrow 0 } \frac { \ln ( \sin 3 x ) } { \ln ( \sin x ) }$$ is equal to

If $$y = \frac{{\sin x}}{{1 + \cos x}},$$ then $$\frac{{dy}}{{dx}}$$ is equal to 

$$\underset { x\rightarrow \infty  }{ Lt } { 5 }^{ x }sin\left( \cfrac { a }{ { 5 }^{ x } }  \right) =$$

$$L\underset { x\rightarrow 0 }{ im } \frac { \sec { 4x-\sec { 2x }  }  }{ \sec { 3x-\sec { x }  }  }=$$

$$\cfrac { d }{ dx } \left( \sin ^{ 5 }{ x } \sin { 5x }  \right) =$$