Given $f(x) = \begin{cases} cx + 1, & x \leq 3 \\ dx + 3, & x > 3 \end{cases}$. If $f$ is continuous at $x = 3$,then $d - c =$ . . . . . . .

The function $f(x) = x + |x|$ is continuous for:

The value of $f(0)$,so that the function $f(x) = \frac{(27 - 2x)^{1/3} - 3}{9 - 3(243 + 5x)^{1/5}}, (x \ne 0)$ is continuous,is given by

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 $f(x) = \left(\frac{1+\tan x}{1+\sin x}\right)^{\operatorname{cosec} x}$ is continuous at $x=0$,then $f(0)$ is equal to

Let $[t]$ denote the greatest integer $\leq t$ and ${t}$ denote the fractional part of $t$. Then the integral value of $\alpha$ for which the left-hand limit of the function $f(x)=[1+x]+\frac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$ at $x=0$ is equal to $\alpha-\frac{4}{3}$ is