If $\lim_{x \to \infty} [\frac{x^2 + x + 1}{x + 1} - ax - b] = 3$, then $a - b = $

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a}-1) x^2+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$ where $a > -1$. Then $\lim _{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _{a \rightarrow 0^{+}} \beta(a)$ are

If $\lim _{x \rightarrow 3} \left( \frac{x^n - 3^n}{x - 3} \right) = 108$ and $n \in N$,then the value of $n$ is

If $\lim_{x \to 2} \frac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1)\log_e(x-1)} = m$, then $a+b+m$ is equal to:

If the function $f(x)$ satisfies $\mathop {\lim }\limits_{x \to 1} \frac{f(x)-2}{x^{2}-1}=\pi,$ evaluate $\mathop {\lim }\limits_{x \to 1} f(x).$

If $\lim _{x \rightarrow 0} \frac{[(a-n) n x-\tan x] \sin n x}{x^2}=0, (n \neq 0)$,then the minimum possible positive value of $a$ is