Limits and Deri...

If $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^{2} + bx + c = 0$$ then $$\displaystyle \lim_{x\rightarrow \dfrac {1}{\alpha}} \sqrt {\dfrac {1 - \cos^{2} (cx^{2} + bx + a)}{4(1 - \alpha x)^{2}}}$$

The value of $$\begin{matrix} lim \\ x\rightarrow \frac { 1 }{ \sqrt { 2 }  }  \end{matrix}\dfrac { x-cos\left( { sin }^{ -1 }x \right)  }{ 1-tan\left( { sin }^{ -1 }x \right)  } is$$

If $$x = 3\cos \theta - 2\cos^{3} \theta$$ and $$y = 3\sin \theta - 2\sin^{3}\theta$$, then $$\dfrac {dy}{dx} =$$

the value of $$\underset { x\longrightarrow \infty  }{ lim } \frac { { X }^{ 4 }sin\left( \frac { 1 }{ x }  \right) +{ x }^{ 3 } }{ 1+\left| x \right| ^{ 3 } } $$

$$\underset { x\rightarrow 1 }{ lim } { \left[ cosec { \dfrac { \pi x }{ 2 }  }  \right]  }^{ { 1 }/{ \left( 1-x \right)  } }$$ (where $$[.]$$ represents the greatest integer function) is equal to

$$\displaystyle \lim_{x\rightarrow 0} \dfrac {\sin 2x + 3x}{2x + \sin 3x}$$ is equal to

The value of $$\underset{x\rightarrow 1}{lim}(2-x)^{tan\left(\dfrac{\pi x}{2}\right)}$$ is

$$\underset{x \rightarrow 1} {lim}\dfrac{x^2-1}{\sin^2x+\cos x\cos (x+2)-\cos^2(x+1)}$$ is-

$$\lim_{x\rightarrow 1}\frac{1-x^{-2/3}}{1-x^{-1/3}}$$

$$\overset {lim}{x \rightarrow \pi/2} \dfrac{\sin(x \ cos x)}{cos(x\, \ sin x)}$$ is equal to