If the function $f(x) = \begin{cases} \frac{(e^{kx} - 1) \tan kx}{4x^2}, & x \neq 0 \\ 16, & x = 0 \end{cases}$ is continuous at $x = 0$,then $k = . . . . . .$.

$A$ function $y=f(x)$ with $f(-1)=-249$ has no maximum and has only one minimum at $x=5$ with $f(5)=75$. Which one of the following is true?

If $f(x) = \begin{cases} [x] + [-x], & x \ne 2 \\ \lambda, & x = 2 \end{cases},$ then $f$ is continuous at $x = 2,$ provided $\lambda$ is (where $[.]$ is the Greatest Integer Function).

On the interval $I = [-2, 2]$,the function $f(x) = \begin{cases} (x + 1) e^{-\left[ \frac{1}{|x|} + \frac{1}{x} \right]} & x \neq 0 \\ 0 & x = 0 \end{cases}$ is given. Which one of the following does not hold good?

Let a function $f: R \rightarrow R$ be defined as
$f(x) = \begin{cases} \sin x - e^x & \text{if } x \leq 0 \\ a + [-x] & \text{if } 0 < x < 1 \\ 2x - b & \text{if } x \geq 1 \end{cases}$
where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$,then $(a + b)$ is equal to:

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x + 2}{x^2 + 3 x + 2}, & x \in R - \{-1, -2\} \\ -1, & x = -2 \\ 0, & x = -1 \end{cases}$ then $f$ is continuous on the set