Limits and Deri...

$$ \lim _{ x\rightarrow 1 }{ \dfrac { \sqrt { 1-\cos { 2\left( x-1 \right)  }  }  }{ x-1 }  }$$

If $$f(x) = 2x^9 - 5x^8 + 7x^6 - 15x^4 + 5x + 7$$, then $$\underset{x \rightarrow 0}{\lim} \dfrac{f (1 - \alpha) - f(1)}{\alpha^3 + 3 \alpha}$$ is 

If $$f(x) = 3x^{10} - 7x^8 + 5x^6 - 21x^3 + 3x^2 - 7$$, then $$\underset{a \rightarrow 0}{\lim} \dfrac{f(1 - \alpha) - f(1)}{\alpha^3 + 3 \alpha}$$ is 

$$\displaystyle \lim_{n \rightarrow \infty}\dfrac {1^{2}+2^{2}+3^{2}+....+n^{2}}{n^{3}}$$ is equal to

If $$y=|\cos x|+|\sin x|$$, then $$\dfrac{dy}{dx}$$ at $$x=\dfrac{2\pi}{3}$$ is 

The value of $$\lim_{x\rightarrow o}\dfrac{\sqrt{\dfrac{1}{2}(1-cos 2 x)}}{x}$$

If $$y=|\cos x|+|\sin x|$$, then $$\dfrac {dy}{dx}$$ at $$x=\dfrac {2\pi}{3}$$  is

The value of $$\underset { x\longrightarrow \infty  }{ Lim } \dfrac{d}{dx}\overset { \sqrt { 3 }  }{ \underset { -\sqrt { 3 }  }{ \int }  } \dfrac{r^3}{(r+1)(r-1)}$$dr,is

$$\mathop {\lim }\limits_{x \to 0} \,\dfrac{1}{x}\,{\sin ^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right)$$ is equal to

$$\mathop {\lim }\limits_{x \to 0} \dfrac{1}{x}{\sin ^{ - 1}}\left( {\dfrac{{2x}}{{1 + {x^2}}}} \right)$$ is equal to