The function $f$ defined on $\left(-\frac{1}{3}, \frac{1}{3}\right)$ by $f(x) = \begin{cases} \frac{1}{x} \log \left(\frac{1+3x}{1-2x}\right), & x \neq 0 \\ k, & x=0 \end{cases}$ is continuous at $x=0$. Then the value of $k$ is:

$f(x) = \begin{cases} 3x - 8 & \text{if } x \leq 5 \\ 2k & \text{if } x > 5 \end{cases}$ is continuous,find $k$.

Let $f:[-2,2] \rightarrow \mathbb{R}$ be a continuous function such that $f(x)$ assumes only irrational values. If $f(\sqrt{2})=\sqrt{2},$ then

The function defined by $f(x) = \begin{cases} (x^2 + e^{\frac{1}{2-x}})^{-1} & x \neq 2 \\ k & x = 2 \end{cases}$ is continuous from the right at the point $x = 2$. Then $k$ is equal to:

If $f(x)$,defined below,is continuous at $x = 4$,then find the values of $a$ and $b$ given that $f(x)$ is continuous on the interval $[0, 8]$.
$f(x) = \begin{cases} x^2 + ax + b, & 0 \leq x < 2 \\ 3x + 2, & 2 \leq x \leq 4 \\ 2ax + 5b, & 4 < x \leq 8 \end{cases}$

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} = $