The value of $k$,for which the function $f(x) = \begin{cases} (\frac{4}{5})^{\frac{\tan 4x}{\tan 5x}}, & 0 < x < \frac{\pi}{2} \\ k + \frac{2}{5}, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$,is:

If the function $f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}$,$x \in R$,is continuous at $x = 0$,then $f(0)$ is equal to:

Let the function $f$ be defined by the equation $f(x) = \begin{cases} 3x & \text{if } 0 \le x \le 1 \\ 5 - 3x & \text{if } 1 < x \le 2 \end{cases}$,then:

If $[x]$ denotes the greatest integer not exceeding $x$ and if the function $f$ defined by $f(x)= \begin{cases} \frac{a+2 \cos x}{x^2} & , x < 0 \\ b \tan \frac{\pi}{[x+4]} & , x \geq 0 \end{cases}$ is continuous at $x=0$, then the ordered pair $(a, b)$ is equal to

The number of points in the interval $[2, 4]$, at which the function $f(x) = [x^2 - x - 1/2]$, where $[·]$ denotes the greatest integer function, is discontinuous, is ————

Let $f: R \rightarrow R$ be defined by $f(x) = \begin{cases} a - \frac{\sin [x-1]}{x-1} & \text{if } x > 1 \\ 1 & \text{if } x = 1 \\ b - \left[ \frac{\sin [x-1] - [x-1]}{([x-1])^3} \right] & \text{if } x < 1 \end{cases}$ where $[t]$ denotes the greatest integer less than or equal to $t$. If $f$ is continuous at $x = 1$,then $a + b =$