The $\mathop {\lim }\limits_{x \to 0} {(\cos x)^{\cot x}}$ is

If $\alpha_1, \alpha_2, \ldots, \alpha_n$ are the roots of $x^n+px+q=0$,then $(\alpha_n-\alpha_1)(\alpha_n-\alpha_2) \ldots (\alpha_n-\alpha_{n-1})=$

Let $f(2) = 4$ and $f'(2) = 4$,then $\mathop {\lim }\limits_{x \to 2} \,\frac{{xf(2) - 2f(x)}}{{x - 2}}$ equals

Let $f(1) = g(1) = k$ and their $n^{th}$ derivatives $f^{(n)}(1), g^{(n)}(1)$ exist and are not equal for some $n$. If $\lim _{x \rightarrow 1} \frac{f(1) g(x) - f(1) - g(1) f(x) + g(1)}{g(x) - f(x)} = 4$,then the value of $k$ is:

The value of $\lim _{x \rightarrow 1} \frac{x^4-\sqrt{x}}{\sqrt{x}-1}$ is

$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\sin x}} - 1}}{x} = $