Let $f$ and $g$ be real-valued functions. If $\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{[f(x)+7]^{2 / 3}}=\frac{7}{4}$, $\lim _{x \rightarrow 0} f(x)=1$ and $\lim _{x \rightarrow 0} g(x)=\alpha$, then $h(x)= \begin{cases} \sin (\alpha x), & 0 \leq x \leq \frac{\pi}{10} \\ \cos (2 \alpha x), & \frac{\pi}{10} < x \leq \frac{\pi}{5} \end{cases}$ is:

The set of values of $x$ for which the function $f(x) = \log \left(\frac{x-1}{x+2}\right)$ is continuous, is

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 $-\frac{1}{512}$

If $f(x) = \begin{cases} \log(\sec^2 x)^{\cot^2 x}, & x \neq 0 \\ K, & x = 0 \end{cases}$ is continuous at $x = 0$,then $K$ is

The function $f(x) = |x| + \frac{|x|}{x}$ is

$f(x) = \begin{cases} \frac{\log x}{x-1}, & \text{if } x \neq 1 \\ k, & \text{if } x=1 \end{cases}$ is continuous at $x=1$,then the value of $k$ is