If $f(x)=\begin{cases}\frac{\sin(a-b)x}{x}, & x<0\\ 2a+b, & x=0\\ \frac{-\tan 12x}{bx}, & x>0\end{cases}$, $(b>0)$ is continuous at $x=0$, then

Match List I with List II

Choose the correct answer from the options given below:

For the function $f(x) = \sqrt{x}$, which of the following statements are true?

(A) $f(x)$ is continuous for $x \geq 0$.

(B) $f(x)$ is continuous for $x \in R$ (Set of real numbers).

(C) $f(x)$ is differentiable for $x > 0$.

(D) $f(x)$ is differentiable for $x \in R$ (Set of real numbers).

Choose the correct answer from the options given below:

If $x = a\cos\alpha + b\sin\alpha$ and $y = a\sin\alpha - b\cos\alpha$, then $\left(x\frac{dy}{dx} - y^2\frac{d^2y}{dx^2}\right)$ is equal to:

Derivative of $x^3 + 1$ with respect to $x^2 + 1$ is

The greatest integer function $f(x) = [x]$ is differentiable for all values of

The derivative of $(\log x)^x$ with respect to $\log x$ is