Limits and Deri...

The value of $$ \lim _ { x \rightarrow 0 } \dfrac { \sin^3 ( \sqrt { x } ) \ln ( 1 + 3 x ) } { \left( \tan ^ { - 1 } \sqrt { x } \right) ^ { 2 } \left( e ^ { 5 ( \sqrt { x } ) } - 1 \right)x } $$ is equal to

The value of $$\lim _{ x\rightarrow \frac { \pi  }{ 2 }  }{ \tan ^{ 2 }{ \left( \sqrt { 2\sin ^{ 2 }{ x } +3\sin { x } +4 } -\sqrt { \sin ^{ 2 }{ x } +6\sin { x } +2 }  \right)  }  } $$ is equal to

Value of $$\int _{ 0 }^{ \pi /2 }{ cos3t } $$ dt is 

$$\mathop {Lt}\limits_{x \to 1} {\left( {1 - x} \right)^{\tan \pi x}} = $$

$$\begin{matrix} lim \\ x\rightarrow a \end{matrix}(2-\frac { a }{ x } ){  }^{ tan(\frac { \pi x }{ 2a } ) }$$ is equal to 

$$\cfrac { d }{ dx } \left[ \sin ^{ 2 }{ \left( \cos ^{ -1 }{ \sqrt { \cfrac { 1+x }{ 1-x }  }  }  \right)  }  \right] =$$

Let $$a \in \left( 0 , \frac { \pi } { 2 } \right)$$, then the value of
$$ \lim _ { a \rightarrow 0 } \frac { 1 } { a ^ { 3 } } \int _ { 0 } ^ { a } \ell n (1+tan a tan x)dx$$ is equal to 

$$ \lim _ { x \rightarrow 0 } \frac { \sin x } { x } = y $$

Let x be an irrational, then $$\underset { m\rightarrow \infty  }{ lim } \underset { n\rightarrow \infty  }{ lim } \left\{ cos(n!\pi x) \right\} ^{ 2m }$$ equals

$$\lim_{x\rightarrow 0}\frac{1-cos^{3}x}{x sib x cos x }$$  is equal to