If a function $f(x) = \begin{cases} \frac{\tan (\alpha + 1)x + \tan 2x}{x}, & \text{if } x > 0 \\ \beta, & \text{at } x = 0 \\ \frac{\sin 3x - \tan 3x}{x^{3}}, & \text{if } x < 0 \end{cases}$ is continuous at $x = 0$,then $|\alpha| + |\beta| =$

If $f(x) = \begin{cases} x e^{-\left( \frac{1}{|x|} + \frac{1}{x} \right)}, & x \ne 0 \\ 0, & x = 0 \end{cases}$,then $f(x)$ is

If the function $f(\alpha) = \begin{cases} \frac{1-\cos 6 \alpha}{36 \alpha^2}, & \alpha \neq 0 \\ k, & \alpha=0 \end{cases}$ is continuous at $\alpha=0$,then $k$ is equal to . . . . . . .

Let the function $f(x) = \begin{cases} \frac{\log_{e}(1+5x) - \log_{e}(1+\alpha x)}{x} & \text{if } x \neq 0 \\ 10 & \text{if } x = 0 \end{cases}$ be continuous at $x = 0$. The value of $\alpha$ is equal to:

If $f(x) = \frac{x^2 - bx + 25}{x^2 - 7x + 10}$ for $x \neq 5$ and $f$ is continuous at $x = 5$,then the value of $f(5)$ is:

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} 2 \sin \left(-\frac{\pi x}{2}\right), & \text{if } x < -1 \\ |ax^2 + x + b|, & \text{if } -1 \leq x \leq 1 \\ \sin(\pi x), & \text{if } x > 1 \end{cases}$. If $f(x)$ is continuous on $R$,then $a + b$ equals ..... .