Limits and Deri...

lf $$f(x)=\displaystyle \frac{x}{\sqrt{1-x^{2}}},g(x)=\frac{x}{\sqrt{1+x^{2}}}$$, then $$\displaystyle \frac{d}{dx}(fog (x))=$$

The integer $$n$$ for which $$\displaystyle \lim_{x\rightarrow 0}\frac{(\cos x-1)(\cos x-e^{x})}{x^{n}}$$ is finite non zero number is

$$\displaystyle \lim_{x\rightarrow 0}\frac{1}{x}\cos ^{ -1 }{ \left( \frac { 1-x^{ 2 } }{ 1+x^{ 2 } }  \right)  } =$$

The right-hand limit of the function $$\sec{x}$$ at $$\displaystyle x=-\frac { \pi  }{ 2 } $$ is

$$\displaystyle \lim_{x\rightarrow 1}(2-x)^{\displaystyle \tan( \frac{\pi x}{2})}=$$

If $$f(x)=(ax+b)\cos x + (cx+d)\sin x$$ and $$f^{'}(x)=x \cos x$$, for all values of $$x\in R$$, then $$a,b,c,d$$ are given by

If $$\displaystyle f(x) = \sqrt {\frac{{x - \sin x}}{{x + {{\cos }^2}x}}} $$ then $$\mathop {\lim }\limits_{x \to \infty } f(x)$$  is

$$\displaystyle\lim_{x \rightarrow \infty}(1^x + 2^x + 3^x+.........+n^x)^{1/x}$$ is

Assertion (A): $$\mathrm{f}(\mathrm{x})=\sin(\pi[x])$$ is differentiable every where $$[\ ]$$ is greatest integer function

$$\displaystyle \lim_{x\to1}{\displaystyle \frac{1-x^2}{\sin 2\pi x}}$$ is equal to