If $f(x) = \begin{cases} \frac{81^{x}-9^{x}}{k^{x}-1} & x \neq 0 \\ 2 & x = 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is

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

If the function $f(x) = \begin{cases} \frac{1}{x} \log_{e}\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right), & x < 0 \\ k, & x = 0 \\ \frac{\cos^{2} x - \sin^{2} x - 1}{\sqrt{x^{2}+1}-1}, & x > 0 \end{cases}$ is continuous at $x = 0$,then $\frac{1}{a} + \frac{1}{b} + \frac{4}{k}$ is equal to:

Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4x](x-\frac{1}{4})^2(x-\frac{1}{2})$,where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
$(A)$ The function $f$ is discontinuous exactly at one point in $(0,1)$
$(B)$ There is exactly one point in $(0,1)$ at which the function $f$ is continuous but $NOT$ differentiable
$(C)$ The function $f$ is $NOT$ differentiable at more than three points in $(0,1)$
$(D)$ The minimum value of the function $f$ is $-\frac{1}{512}$

The function $f(x)=\frac{\tan \{\pi[x-\frac{\pi}{2}]\}}{2+[x]^{2}}$,where $[x]$ denotes the greatest integer $\leq x$,is

If $\begin{aligned} f(x) &= \frac{4 \sin \pi x}{5 x} \text{ for } x \neq 0 \\ &= 2k \text{ for } x = 0 \end{aligned}$ is continuous at $x = 0$,then the value of $k$ is