Limits and Deri...

The limit of $$\left\{\frac{1}{x}\sqrt{1-x}-\sqrt{1+\frac{1}{x^2}}\right\}$$ as $$x\rightarrow 0$$

If $$y = \tan^{-1} \left (\dfrac {1}{1 + x + x^{2}}\right ) + \tan^{-1} \left (\dfrac {1}{x^{2} + 3x + 2}\right ) + \tan^{-1} \left (\dfrac {1}{x^{2} + 5x + 6}\right ) + .... +$$ upto $$n$$ terms then $$\dfrac {dy}{dx}$$ at $$x = 0$$ and $$n = 1$$ is equal to

What is $$\displaystyle \lim_{x \rightarrow 0 }  x^2 \sin \left(\frac{1}{x}\right)$$ equal to ? 

If $$f\left( x \right) =\begin{vmatrix} \sin { x }  & \cos { x }  & \tan { x }  \\ { x }^{ 3 } & { x }^{ 2 } & x \\ 2x & 1 & 1 \end{vmatrix}$$, then $$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { f\left( x \right)  }{ { x }^{ 2 } }  } $$ is

$$\displaystyle\lim_{x\rightarrow \frac{\pi}{2}}(\pi - 2x^{\cos x})$$ is equal to :

The derivative of $$ln(x+\sin x)$$ with respect to $$(x+\cos x)$$ is

Let   $$C(\theta)=\displaystyle\sum _{n=0}^{\infty}\dfrac{\cos(n\theta)}{n!}$$
Which of the following statements is FALSE? 

$$\displaystyle\lim_{x\rightarrow\frac{\pi}{6}}\frac{\sin\left(x-\displaystyle\frac{\pi}{6}\right)}{\sqrt{3-2cos x}}$$ is equal to :

$$\displaystyle \lim_{x\rightarrow \pi/4} \dfrac {\tan x - 1}{\cos 2x}$$ is equal to

The value of $$\displaystyle \lim _{ x\rightarrow \pi /6 }{ \cfrac { 2\sin ^{ 2 }{ x } +\sin { x } -1 }{ 2\sin ^{ 2 }{ x } -3\sin { x } -1 }  } $$ is