The set of values of $x$ for which the function $f(x) = \log \left(\frac{x-1}{x+2}\right)$ is continuous,is

If $f(x) = \begin{cases} \frac{x^4 - 16}{x - 2}, & x \neq 2 \\ 16, & x = 2 \end{cases}$,then:

Statement-$1$: The equation $x \log x = 2 - x$ is satisfied by at least one value of $x$ lying between $1$ and $2$.
Statement-$2$: The function $f(x) = x \log x$ is an increasing function in $[1, 2]$ and $g(x) = 2 - x$ is a decreasing function in $[1, 2]$,and the graphs represented by these functions intersect at a point in $[1, 2]$.

If $f(x) = \begin{cases} x, & 0 \le x \le 1 \\ 2x - 1, & x > 1 \end{cases}$,then

Examine the continuity of the function $f(x) = 2x^{2} - 1$ at $x = 3$.

The value of $f(0)$ so that the function $f(x) = \frac{2^x - 2^{-x}}{x}$ for $x \neq 0$ is continuous at $x = 0$ is: