If $\lim_{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+\ldots+(nk+n)] = 33 \cdot \lim_{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot [1^k + 2^k + 3^k + \ldots + n^k]$,then the integral value of $k$ is equal to $....$

$\lim _{x \rightarrow 0} \frac{1-\cos x \cos 2 x}{\sin ^2 x} = $

$\mathop {Limit}\limits_{x \to \frac{\pi }{2}} \,\frac{{\sin x}}{{{{\cos }^{ - 1}}\left[ {\frac{1}{4}\,(3\sin x\, - \,\sin 3x)} \right]}}\,$,where $[ \cdot ]$ denotes the greatest integer function,is

Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos ^{-1}\left(1-\{x\}^2\right) \sin ^{-1}(1-\{x\})}{\{x\}-\{x\}^3}, x \neq 0$. If $L$ and $R$ respectively denote the left-hand limit and the right-hand limit of $f(x)$ at $x=0$,then $\frac{32}{\pi^2}\left(L^2+R^2\right)$ is equal to:

If $0 < p < q$,then $\lim _{n \rightarrow \infty}\left(q^n+p^n\right)^{1 / n}$ is equal to :

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 3} \frac{x^{4}-81}{2 x^{2}-5 x-3}$