If $f(x) = \begin{cases} \frac{1-\sqrt{2} \sin x}{\pi-4x} & \text{if } x \neq \frac{\pi}{4} \\ a & \text{if } x = \frac{\pi}{4} \end{cases}$ is continuous at $x = \frac{\pi}{4}$,then $a$ is equal to

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 $f(x)$ is continuous and $f\left( \frac{9}{2} \right) = \frac{2}{9}$, then $\lim_{x \to 0} f \left( \frac{1 - \cos 3x}{x^2} \right)$ is equal to:

Let $f : [a, b] \rightarrow [1, \infty)$ be a continuous function and let $g : \mathbb{R} \rightarrow \mathbb{R}$ be defined as $g(x) = \begin{cases} 0 & \text{if } x < a \\ \int_a^x f(t) dt & \text{if } a \leq x \leq b \\ \int_a^b f(t) dt & \text{if } x > b \end{cases}$. Then:

Let the function $f$ be defined by $f(x) = \frac{x - |x|}{x}$ for $x \neq 0$ and $f(0) = 2$. Then $f$ is:

If $f$ is defined by $f(x) = \begin{cases} \frac{1-\cos ax}{x \sin x}, & x \neq 0 \\ \frac{1}{2}, & x = 0 \end{cases}$ and $f$ is continuous at $x=0$,then $a^{2} =$ . . . . . . .