Limits and Deri...

If $$f(x)$$ is the integral of $$\dfrac{2\sin{x}-\sin{2x}}{x^{3}},\ x\neq 0$$. Find $$\lim _{ x\rightarrow 0 }{ f^{ ' }\left( x \right)  } $$, where $$f^{ ' }\left( x \right) =\dfrac{df{(x)}}{dx}$$

$$\underset{x\rightarrow 0}{lim} \dfrac{\sqrt{a^2 -ax+x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} -\sqrt{a-x}}$$ is equal to (a > 0)

If $$\dfrac{dx}{tan(x+y)} = \dfrac{dy}{cot(x+y)} = \dfrac{dz}{I}$$ then z in term of x & y can be expressed as

The value of $$\displaystyle lim_{x\to 0} \dfrac{cos (sin x) - cos x}{x^4} $$ is equal to

The derivative of $$\cos^{-1}\dfrac{1-x^{2}}{1+x^{2}} w.r.t\ \tan^{-1}\dfrac{2x}{1-x^{2}}$$ is

$$\displaystyle\lim _{ x\rightarrow \dfrac { x }{ 2 }  }{ \dfrac { \cot { x } -\cos { x }  }{ \left( \pi -2x \right) ^{ 3 } }  } $$ is equal to 

The value of $$\lim_{x \rightarrow 1} \sec \dfrac{\pi}{2x} \log x$$ is-

$$\lim _ { x \rightarrow 1 } \{ 1 - x + [ x + 1 ] + [ 1 - x ] \} , \text { where } [ x ]$$ denotes greatest integer function is

If $$2f(\sin x)+\sqrt {2}f(\cos x)=\tan x,\ (x> 0)$$, then $$\displaystyle \lim _{ x\rightarrow 1 }{ \sqrt { 1-x } f\left( x \right) = }$$ 

$$ \underset { x\rightarrow 0 }{ lim } \cfrac { 1+cos\left( \pi x \right)  }{ \left( 1-x \right)^ 2 }   $$ is equal to :