If $\lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{n}-n}{x-1}=820, (n \in N)$ then the value of $n$ is equal to

If $\lim _{x \rightarrow 0}\left(\frac{\cos 4 x+a \cos 2 x+b}{x^4}\right)$ is finite,then the values of $a, b$ are respectively :

If $\mathop {\lim }\limits_{x \to \infty } \left\{ {\ln \left( {{x^2} + 5x} \right) - 2\ln \left( {cx + 1} \right)} \right\} = -2$,then:

If $\lim _{x \rightarrow 0} \frac{a x^2 e^x - b \log _e(1+x) + c x e^{-x}}{x^2 \sin x} = 1$,then $16(a^2 + b^2 + c^2)$ is equal to ...........................

If $\lim _{x}$ ${\rightarrow \infty} \frac{(\sqrt{2 x+1}+\sqrt{2 x-1})^8+(\sqrt{2 x+1}-\sqrt{2 x-1})^8(P x^4-16)}{(x+\sqrt{x^2-2})^8+(x-\sqrt{x^2-2})^8} = 1$,then $P=$

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) = $