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 $[0, 8]$.
$f(x) = \begin{cases} x^2 + ax + b, & 0 \leq x < 2 \\ 3x + 2, & 2 \leq x \leq 4 \\ 2ax + 5b, & 4 < x \leq 8 \end{cases}$

Let $f: R \rightarrow R$ be a continuous function such that $f(x^2) = f(x^3)$ for all $x \in R$. Consider the following statements:
$I.$ $f$ is an odd function.
$II.$ $f$ is an even function.
$III.$ $f$ is differentiable everywhere.
Then,

If $f(x) = \lim_{n \rightarrow \infty} \left( \frac{\log(2+x) - x^{2n} \sin x}{1+x^{2n}} \right)$ for $0 \leq x \leq \frac{\pi}{2}$, then at $x=1$, $f(x)$ is

Consider the function $f(x)=\begin{cases} \frac{a(7x-12-x^2)}{b|x^2-7x+12|} & , x<3 \\ 2^{\frac{\sin(x-3)}{x-[x]}} & , x>3 \\ b & , x=3 \end{cases}$ where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a, b)$ such that $f(x)$ is continuous at $x=3$,then the number of elements in $S$ is:

In order that the function $f(x) = (x + 1)^{\cot x}$ is continuous at $x = 0$,$f(0)$ must be defined as

If $f(x) = \frac{2 - \sqrt{x + 4}}{\sin 2x}, (x \neq 0),$ is a continuous function at $x = 0$,then $f(0)$ equals: