Limits and Deri...

if $$x=\cos^{3}\theta,y=\sin^{3}\theta$$, then $$\sqrt{1+\left(\dfrac{dy}{dx}\right)^{2}}=$$

$$\displaystyle \lim _{ x\rightarrow 2 }{ \frac { \sqrt [ 3 ]{ 60+{ x }^{ 2 } } -4 }{ \sin { \left( x-2 \right)  }  }  } $$

The value of $$\displaystyle \lim _{ x\rightarrow 0 }{ \csc^{ 4 }{ x } \int _{ 0 }^{ { x }^{ 2 } }{ \frac { ln\left( 1+4t \right)  }{ { t }^{ 2 }+1 }  } dt } $$ is 

$$\displaystyle \lim_{x\rightarrow 0^{+}}{(\csc x)^{1/\log x}}$$=

$$\displaystyle \lim_{x\rightarrow \infty}{x^{2}\sin\left(\log_{e}\sqrt{\cos\dfrac{\pi}{x}}\right)}$$

$$ \underset { x\rightarrow \cfrac { \pi  }{ 2 }  }{ lim } \cfrac { cot \,  x-cos\, x }{ \left( \pi -{ 2x } \right)^ 3 } $$ equals

$$\underset { x\rightarrow \frac { \pi  }{ 2 }  }{ lim } \frac { (1-sinx)({ 8x }^{ 2 }-{ \pi  }^{ 3 })cosx }{ { (\pi -2x) }^{ 4 } } $$

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

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 :

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