Limits and Deri...

Let $$\displaystyle x^{cos\,y} + y^{cox\,x} = 5 $$ , Then 

Value of $$L=\displaystyle\lim_{n\rightarrow \infty n}\dfrac{1}{4}\left[1.\left(\displaystyle\sum_{k=1}^{n}k\right)+2.\left(\sum_{k=1}^{n-1}k\right)+3.\left(\sum_{k=1}^{n-2}k\right)+...+n.1\right]$$ is

Solution of the equation $$ \dfrac {dy }{dx}  + \dfrac {1}{x} \tan y = \dfrac {1}{x^2}  \tan y \sin y$$ is 

$$\displaystyle \lim _{x \rightarrow 1}\left(\dfrac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\dfrac{1-\cos (x+1)}{(x+1)^{2}}} $$ is equal to:

The value of $$\displaystyle \lim_{n\infty}\dfrac{1}{n^2}\left\{ sin^3\dfrac{\pi}{4n}+2sin^3\dfrac{2\pi}{4n} + ... + nsin^3\dfrac{n\pi}{4n}\right\}$$ is equal to 

If $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{x^{n}-\sin x^{n}}{x-\sin ^{n} x} $$ is non-zero finite, then $$ n $$ must be equal

If $$ L=\displaystyle \lim _{x \rightarrow 0} \dfrac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}}=\infty $$

Equation $$ a x^{2}+b x+c=0 $$ has

$$\displaystyle \lim _{x \rightarrow \infty} \dfrac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}} $$ is equal to

$$\displaystyle  \lim _{x \rightarrow \pi / 2} \dfrac{\sin (x \cos x)}{\cos (x \sin x)} $$ is equal to

The value of $$ \displaystyle \lim _{x \rightarrow 0} \dfrac{\sqrt{\dfrac{1}{2}(1-\cos 2 x)}}{x} $$ is