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