If $f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x}, (x \neq 0),$ is a continuous function at $x = 0$,then $f(0)$ equals:

Let $[t]$ represent the greatest integer not more than $t$. Then the number of discontinuous points of $f(x) = [x^{1/x}]$ in $(0, \infty)$ is

For $a, b > 0$,let $f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0 \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b \sqrt{a} x \sqrt{x}}, & x > 0 \end{cases}$ be a continuous function at $x = 0$. Then $\frac{b}{a}$ is equal to

If $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is

Let $f$ be a continuous,periodic even function defined on $\mathbb{R}$ such that $f(0) = 1$,$f(2) = -1$ and the period of $f$ is $4$. The minimum number of roots of the equation $f(x) = 0$ in the interval $[-10, 10]$ will be:

If $[x]$ is the greatest integer function and $f(x) = \begin{cases} 2[x] - \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$ is a real-valued function,then $f$ is