Limits and Deri...

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

Let  $$f ( \beta ) = \lim _ { \alpha \rightarrow \beta } \dfrac { \sin ^ { 2 } \alpha - \sin ^ { 2 } \beta } { \alpha ^ { 2 } - \beta ^ { 2 } },$$  then  $$f \left( \dfrac { \pi } { 4 } \right)$$  is greater than-

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

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

The value of $$\begin{matrix} lim \\ x\rightarrow 0 \end{matrix}$$ $$[\frac{x}{sinx}],$$ where [.] represents the greatest inter function , is 

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

The value of $$\displaystyle \lim_{x\rightarrow 0}\left(\dfrac {1}{x^{2}}-\cot x\right)$$ equals

$$\underset { x\rightarrow 0 }{ Lt\quad  } \frac { sec\quad x-1 }{ { \left( sec\quad x+\quad 1 \right)  }^{ 2 } } =$$

If $$u=f(x^{2}), v=g(x^{3}),f(x)=sinx, g^{1}(x)=cosx$$ then find $$\frac{du}{dv}$$

If$$y=sin x\left[ \dfrac { 1 }{ sin x sin 2 x } +\dfrac { 1 }{ sin 2 x.sin 3 x } +.....+\dfrac { 1 }{ sinn x sin\left( n+1 \right) x }  \right]$$ then $$\dfrac { dy }{ dx } =$$