If $f(x) = \frac{\log_e(1 + x^2 \tan x)}{\sin x^3}, x \neq 0$ is to be continuous at $x = 0$,then $f(0)$ must be equal to

If $f(x) = \begin{cases} \frac{\sqrt{\pi} - \sqrt{\cos^{-1} x}}{\sqrt{x+1}}, & x \neq -1 \\ \frac{1}{\sqrt{\lambda \pi}}, & x = -1 \end{cases}$ is right continuous at $x = -1$,then $\lambda = $

Let $f:R \rightarrow R$ be defined as $f(x) = \begin{cases} 0, & x \text{ is irrational} \\ \sin |x|, & x \text{ is rational} \end{cases}$. Then,which of the following is true?

The function $f(x) = [x] \cos \left( \frac{2x - 1}{2} \pi \right)$,where $[.]$ denotes the greatest integer function,is discontinuous at

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

If the function given by $f(x) = \left(\frac{4x+1}{1-4x}\right)^{\frac{1}{x}}$ for $x \neq 0$ is continuous at $x = 0$,then the value of $f(0)$ is