Limits and Cont...

$\displaystyle\lim _{ x\rightarrow 0 }{ \dfrac { { \left( \cos { x }  \right)  }^{ 1/2 }-{ \left( \cos { x }  \right)  }^{ 1/3 } }{ \sin ^{ 2 }{ x }  }  }$ is $$

$\displaystyle \lim _{ x\rightarrow 0 }{ \frac { x\left( { e }^{ \sin { x }  }-1 \right)  }{ 1-\cos { x }  }  }$

$\underset { x\rightarrow 0 }{ lim } (\cos  x+a\sin  b{ x) }^{ \frac { 1 }{ x }  }$ is equal to 

$\displaystyle \lim_{x\rightarrow 0}{\dfrac{x(1+a\cos x)-b\sin x}{x^{3}}}=1$ then

The value of $\displaystyle\lim_{\theta \rightarrow 0^+}\dfrac{\sin\sqrt{\theta}}{\sqrt{\sin\theta}}$ is equal to?

$\displaystyle\lim_{x\rightarrow 0}\left(\dfrac{1}{\sin^2x}-\dfrac{1}{\sin h^2x}\right)=?$

Let $f$ : $\left ( \dfrac{\pi}{2}, \dfrac{\pi}{2} \right )\rightarrow R$ , $f(x) = \left \{\begin{matrix} \lim_{n\rightarrow \infty }\dfrac{(tanx)^{2n} + x^2}{sin^2x + (tanx)^{2n}}; & x \neq 0 \\ 1; & x = 0 \end{matrix} \right., n \in N$ . Which of the following holds good ?

If $\lim _{x \rightarrow 0}\left(\cos x+a^{3} \sin \left(b^{6} x\right)\right)^{\frac{1}{x}}=e^{512}$ then value of $ab^2$ is equal to

$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sin(\pi \cos^2x)}{x^2}$ is equal to?

The value of $\begin{matrix} lim \\ x\rightarrow y \end{matrix}\dfrac { { sin }^{ 2 }x-sin^{ 2 }y }{ { x }^{ 2 }-{ y }^{ 2 } }$