Limits and Cont...

Let [x] denote the greatest integer less than or equal to x. Then :
$$\underset { x\rightarrow 0 }{ lim } \dfrac { tan\left( \pi { sin }^{ 2 }x \right) +\left( \left| x \right| -sin \right) \left( x\left[ x \right]  \right) ^{ 2 } }{ { x }^{ 2 } } :$$

$$\underset { x\rightarrow \frac { \pi  }{ 2 }  }{ lim } \frac { cotx-cosx }{ { (\pi -2x) }^{ 3 } } $$ equals :

$$\displaystyle\lim _{ n\rightarrow \infty  }{ { n }^{ 2 }\left( { x }^{ \dfrac { 1 }{ n }  }-{ x }^{ \dfrac { 1 }{ n+1 }  } \right) ,x>0 } $$ is equal to 

If $$\displaystyle \lim _{ x\rightarrow 0 }[1+ax+bx^{2}]^{(2/x)}=e^{3}$$, then 

If [.] deotes the greatest integer function then
$$\begin{matrix} lim \\ x\rightarrow \pi /2 \end{matrix}\left[ \frac { x-\frac { \pi  }{ 2 }  }{ cosx }  \right] $$ is equal to

$$\displaystyle x\xrightarrow { lim } a\left(\sin\frac{x-a}{2}\tan\frac{\pi x}{2a}  \right) $$

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

$$\underset { x\rightarrow 0 }{ lim } \frac { sin({ 6x }^{ 2 }) }{ Incos({ 2x }^{ 2 }-x) } =$$

$$\underset { x\rightarrow -\infty  }{ lim } \frac { ({ 3x }^{ 4 }+{ 2x }^{ 2 })sin(\frac { 1 }{ x } )+{ |x| }^{ 3 }+5 }{ { |x }|^{ 3 }+{ |x| }^{ 2 }+|x|+1 } =$$

The value of $$\displaystyle \lim_{x \rightarrow 0} (\sin x)^{\dfrac{1}{x}}+(1+x)^{(\sin x)})=0$$, where $$x > 0$$, is :