Limits and Cont...

$$\lim_{n\rightarrow \infty}\dfrac{1}{n^{2}}\left[\sin^{3}\dfrac{\pi}{4n}+2\sin^{3}\dfrac{2\pi}{4n}+3\sin^{3}\dfrac{3\pi}{4n}+....+n\sin^{3}\dfrac{n\pi}{4n}\right]=$$

If $$ \alpha \quad and \beta $$ are the roots of the equation  $$ {ax}^{2}+bx+c=0 $$, then 
 $$ \underset { x\rightarrow \cfrac { \pi  }{ 2 }  }{ lim } \cfrac { tan\left[ \left( \alpha +\beta  \right) x \right]  }{ sin\left[ \left( \alpha \beta  \right) x \right]  }  $$ is equal to :

$$\begin{matrix} lim \\ n\rightarrow \infty  \end{matrix}\int _{ 0 }^{ 1 }{ \frac { { nx }^{ n-1 } }{ 1+{ x }^{ 2 } }  } dx=$$

If $$L = \underset{x \rightarrow 0}{lim} \dfrac{a \, sin \, x - sin \, 2x}{tan^3 x}$$ is finite, then the value of L is :

$$\underset { x\rightarrow 0 }{ lim } \left( \dfrac { \left( 1+x \right) ^{ \dfrac { 1 }{ x }  } }{ e }  \right) ^{ \dfrac { 1 }{ sinx }  }$$ is equal to 

$$ \underset { x\rightarrow 0 }{ lim } \cfrac { { \left( 25 \right)  }^{ x }-2\left( 15 \right)^ x+{ 9 }^{ x } }{ cos6x-cos2x }  $$ is equal to :

$$Lt_{x\to 0} \dfrac{sin x - x+\dfrac{x^3}{6}}{x^5}=$$_________

$${ lim }_{ x\rightarrow 1 }(1+cos\pi ){ cot }^{ 2 }\pi x=-----$$

$$\displaystyle \lim _{ \theta \rightarrow \pi /2 }{ \dfrac { 1-\sin  \theta  }{ (\pi /2-\theta )\cos { \theta  }  }  } $$ is equal to

If $$\underset {x \rightarrow \infty}{lim} (1+\frac {a}{x}-\frac {4}{x^{2}})^{2x} =e^{3}$$ , then 'a' is equal to :