Limits and Deri...

If $\displaystyle \lim_{x\rightarrow 0}(f(x)\:g(x))$ exists for any functions $f$ and $g$ then

$\displaystyle \lim_{x\rightarrow\infty}\left(\frac{\sqrt{(1 - \cos x)+ \sqrt{(1 - \cos x)+ \sqrt(1 - \cos x)+...\infty) - 1}}}{x^2}\right)$ equals to

$\underset{x\rightarrow0}{lim}\displaystyle\frac{1-cos^{3}x+sin^{3}x+\ell n(1+x^{3})+\ell n(1+cos\,\,x)}{x^{2}-1+2\,cos^{2}x+tan^{4}x+sin^{3}x}$ is equal to -

let a, b, c are non zero constant number then $\lim_{r\rightarrow\infty}\displaystyle\frac{cos\displaystyle\frac{a}{r}-cos\displaystyle\frac{b}{r}cos\displaystyle\frac{c}{r}}{sin\displaystyle\frac{b}{r}sin\displaystyle\frac{c}{r}}$ equals to

$\displaystyle f^{ ' }\left( x \right) =g\left( x \right)$ and $\displaystyle g^{ ' }\left( x \right) =-f\left( x \right)$ for all real x and $\displaystyle f\left( 5 \right) =2=f^{ ' }\left( 5 \right)$ then $\displaystyle f^{ 2 }\left( 10 \right) +g^{ 2 }\left( 10 \right)$ is -

Evaluate $\displaystyle \lim_{n\rightarrow \infty }\left [ \frac{n!}{n^{n}} \right ]^{1/n}$.

$\displaystyle \frac{d}{dx}\left ( \tan ^{-1}\left ( \frac{a-x}{1+ax} \right ) \right )$ equals if ax > -1

If $f(x) = \displaystyle \left | \cos x-\sin x \right |$ then $\displaystyle f'\left ( \dfrac{\pi}4 \right )$ is equal to-

If $\displaystyle y=\frac{1}{1+x^{\beta -\alpha}+x^{\gamma -\alpha}}+\frac{1}{1+x^{\alpha-\beta}+x^{\gamma -\beta }}+\frac{1}{1+x^{\alpha -\gamma }+x^{\beta-\gamma }}$
then $\displaystyle \frac{dy}{dx}$ is equal to-

If ${ S }_{ n }$ denotes the sum of $n$ terms of $g.p$. whose common ratio is $r$, then $\displaystyle \left( r-1 \right) \frac { d{ S }_{ n } }{ dr }$ is equal to