If $\mathop {\lim }\limits_{x \to 2} \frac{{\tan (x - 2)({x^2} + (a - 2)x - 2a)}}{{({x^2} - 4x + 4)}} = 7$,then $a$ is -

If $\alpha, \beta$ are the roots of the equation $ax^2 + bx + c = 0$,then $\lim_{x \rightarrow \beta} \frac{1 - \cos(ax^2 + bx + c)}{(x - \beta)^2}$ is

If $\alpha$ is the positive root of the equation $p(x) = x^{2} - x - 2 = 0$,then $\lim_{x \rightarrow \alpha^{+}} \frac{\sqrt{1 - \cos(p(x))}}{x + \alpha - 4}$ is equal to

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 $\lim _{x \rightarrow 1} \frac{x^2-ax+b}{x-1}=5$,then $(a+b)$ is equal to

If $\mathop {\lim }\limits_{x \to 0} \frac{{[(a - n)nx - \tan x]\sin nx}}{{{x^2}}} = 0,$ where $n$ is a non-zero real number,then $a$ is equal to