Limits and Cont...

$$\lim\limits_{x\rightarrow0}\dfrac{x\tan 2x-2x\tan x}{\left(1-\cos 2x\right)^{2}}=$$

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

$$\lim_{x\to \infty} \dfrac{\sqrt{x^2 + sin^2x}}{x+cosx}$$ equals

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

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

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

$$\displaystyle\lim_{h\rightarrow 0}\dfrac{\sin\sqrt{x+h}-\sin\sqrt{x}}{h}=$$__________.

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

Evaluate: $$\underset{x\rightarrow 0}{lim} \dfrac{e^{1/x} - 1}{e^{1/x }+ 1}$$

If $$\underset { x\rightarrow { 0 } }{ lim } \dfrac { \left( ax+b \right) -\sqrt { 4+\sin x }  }{ \tan\quad x } =\dfrac { 27 }{ 4 } ~where ~a,b\in R$$ then the value of