If the function $f(x)$ satisfies $\lim_{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1} = \pi$,then $\lim_{x \rightarrow 1} f(x) = $

If $\mathop {\lim }\limits_{x \to 1} \frac{{{x^4} - 1}}{{x - 1}} = \mathop {\lim }\limits_{x \to k} \frac{{{x^3} - {k^3}}}{{{x^2} - {k^2}}}$,then $k$ is

The integral value of $n$ for which $\lim _{x \rightarrow 0} \frac{(\cos x-1)(\cos x-e^x)}{x^n}$ is a finite non-zero real number is

If $\lim _{x \rightarrow 0} \frac{2 a \sin x-\sin 2 x}{\tan ^{3} x}$ exists and is equal to $1$, then the value of $a$ is

If $\lim _{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^3}=-1$,where $\lambda, \mu \in \mathbb{R}$,then $\lambda+\mu$ is equal to

The largest value of the non-negative integer $a$ for which $\lim _{x \rightarrow 1}\left\{\frac{-a x+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$ is