Limits and Deri...

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

$$if\left( x \right) =\left[ x-3 \right] +\left[ x-4 \right] \quad for\quad x\epsilon R\quad then\quad \underset { x\rightarrow 3 }{ lim } f\left( x \right) =$$

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

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

$$\displaystyle\lim_{x\rightarrow 0}\dfrac{x\tan 2x-2x\tan x}{(1-\cos 2x)^2}$$ is equal to?

If $$f(x)$$ is odd linear polynomial with $$f(1)=1$$, then $$\underset{x \to 0} {\lim} \dfrac{2^{f(\tan x)}-2^{f(\sin x)}}{x^{2}f(\sin x)}$$ is 

 $$\lim _ { x \rightarrow \infty } \dfrac { \sin \left( \pi \cos ^ { -2 } x \right) } { x ^ { 2 } }$$  is equal to:

$$\lim _ { x \rightarrow 0 } \dfrac { \sin 4 x } { \tan 7 x } =$$

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

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