$f(x) = \begin{cases} \frac{1-\cos kx}{x^2}, & x \le 0 \\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & x > 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is:

Consider $f(x) = [x]|x^3 - 2x^2 - x + 2|$ in $[-\frac{3}{2}, \frac{9}{2}]$. The number of points where $f(x)$ is discontinuous is (where $[.]$ denotes the greatest integer function).

If $f(x) = \begin{cases} \frac{1 - [x]}{1 + x}, & x \ne -1 \\ 1, & x = -1 \end{cases}$,then the value of $f(|2k|)$ will be (where $[.]$ denotes the greatest integer function). Which of the following statements is true?

If $f(x) = \begin{cases} \frac{\tan(2p-7)x + \tan 3x}{x}, & x < 0 \\ p-q, & x=0 \\ q\left(\frac{\sqrt{x^2+x}-\sqrt{x}}{x^{3/2}}\right), & x > 0 \end{cases}$. If $f(x)$ is continuous at $x=0$, then $\frac{q}{p} = $

If $f(x) = x^3 + 7x - 1$,then $f(x)$ has a zero between $x = 0$ and $x = 1$. The theorem which best describes this is:

If the function $f(x) = \frac{\sqrt{1+x}-1}{x}$ is continuous at $x=0$,then $f(0) = $