In the interval $(-2 \pi, 0)$,the function $f(x) = \sin \left(\frac{1}{x^3}\right)$

Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \alpha+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & x < 0 \end{cases}$. If $f$ is continuous at $x=0$,find the value of $\alpha + \beta$.

If the function $f(x) = \begin{cases} 3ax + b, & \text{for } x < 1 \\ 11, & \text{for } x = 1 \\ 5ax - 2b, & \text{for } x > 1 \end{cases}$ is continuous at $x = 1$,then the values of $a$ and $b$ are:

If the function $f(x) = \begin{cases} x - \frac{|x|}{x}, & x < 0 \\ x + \frac{|x|}{x}, & x > 0 \\ 1, & x = 0 \end{cases}$,then which of the following is true?

If $f(x) = \operatorname{sgn}((x^2 - kx + 6)(\sin x - 1/2))$ (where $k > 0$) has exactly $4$ points of discontinuity in $(0, 6)$,then the maximum integral value of $k$ is:

If $f(x)= \begin{cases} \frac{x-[x]}{x-2}, & x>2 \\ b, & x=2 \\ \frac{|x^2-x-2|}{a(2+x-x^2)}, & -1 < x \leq 2 \\ 2a-b, & x \leq -1 \end{cases}$ is continuous on $R$,then $\lim _{x \rightarrow 0} \frac{\sin ^2 ax+x \tan bx}{x^2}=$