Limits and Deri...

If $$\displaystyle {f}'(x) = sin\,x + sin\,4x .\, cos \,x $$ then $$\displaystyle {f}'(x) \left (2x^{2} + \dfrac{\pi}{2} \right ) $$ at $$ x = \sqrt{\dfrac{\pi}{2}} $$ is equal to 

If $$\displaystyle sin\, y = x\, sin ( a + y) $$ and
$$\displaystyle \dfrac{dy}{dx} = \dfrac{A}{ 1 + x^{2} - 2x \, cos a } $$ then the value of $$ A $$ is

$$ \displaystyle \lim _{x \rightarrow-\infty} \dfrac{x^{2} \tan \dfrac{1}{x}}{\sqrt{8 x^{2}+7 x+1}} $$ is equal to

$$\displaystyle \lim _{x \rightarrow 0} \dfrac{x\left(e^{x}-1\right)}{1-\cos x} $$ is equal to

$$ \displaystyle \lim _{x \to \pi / 2}\left[x \tan x-\left(\dfrac{\pi}{2}\right) \sec x\right] $$ is equal to 

If $$ f(x)=\dfrac{\cos x}{(1-\sin x)^{1 / 3}}, $$ then

$$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x^{n}}{(\sin x)^{m}},(m<n) $$ is equal to

$$\displaystyle  \lim _{x \rightarrow 1} \dfrac{1+\sin \pi\left(\dfrac{3 x}{1+x^{2}}\right)}{1+\cos \pi x} $$ is equal to

 $$ The \ value \ of  \displaystyle \lim _{x \rightarrow 1}(2-x)^{\tan \dfrac{\pi x}{2}} $$ is

$$\displaystyle \lim _{x \rightarrow 1} \dfrac{1-x^{2}}{\sin 2 \pi x} \text { is equal to }$$