Let [.] be the greatest integer less than or equal to t. Then the least value of p ∈ N for which

$ \lim\limits_{x \to 0^+} \left( x \left[ \frac{1}{x} \right] + \left[ \frac{2}{x} \right] + \ldots + \left[ \frac{p}{x} \right] \right) - x^2 \left( \left[ \frac{1}{x^2} \right] + \left[ \frac{2}{x^2} \right] + \ldots + \left[ \frac{9^2}{x^2} \right] \right) \geq 1 $ is equal to _______.

Let $f(x)=\lim \limits_{n \rightarrow \infty} \sum\limits_{r=0}^n\left(\frac{\tan \left(x / 2^{r+1}\right)+\tan ^3\left(x / 2^{r+1}\right)}{1-\tan ^2\left(x / 2^{r+1}\right)}\right)$ Then $\lim\limits_{x \rightarrow 0} \frac{e^x-e^{f(x)}}{(x-f(x))}$ is equal to ___________.

Let $\mathrm{f}(x)=\left\{\begin{array}{lc}3 x, & x<0 \\ \min \{1+x+[x], x+2[x]\}, & 0 \leq x \leq 2 \\ 5, & x>2\end{array}\right.$

where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not continuous and is not differentiable, respectively, then $\alpha+\beta$ equals _______ .

Let the function,

$$f(x)= \begin{cases}-3 \mathrm{ax}^2-2, & x<1 \\ \mathrm{a}^2+\mathrm{b} x, & x \geqslant 1\end{cases}$$

be differentiable for all $x \in \mathbf{R}$, where $\mathrm{a}>1, \mathrm{~b} \in \mathbf{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta \sqrt{3}, \alpha, \beta \in Z$, then the value of $\alpha+\beta$ is ___________ .