Suppose $f(x) = \begin{cases} a + bx, & x < 1 \\ 4, & x = 1 \\ b - ax, & x > 1 \end{cases}$ and if $\lim_{x \to 1} f(x) = f(1)$,what are the possible values of $a$ and $b$?

The function $f(x) = \begin{cases} \frac{x - |x|}{x}, & x \neq 0 \\ 2, & x = 0 \end{cases}$

If $f(x) = \begin{cases} (1 + 2x)^{1/x}, & x \ne 0 \\ e^2, & x = 0 \end{cases}$,then:

Which of the following function$(s)$ not defined at $x = 0$ has/have an irremovable discontinuity at $x = 0$?

Let $k$ be a non-zero real number. If $f(x) = \begin{cases} \frac{(e^x - 1)^2}{\sin (x/k) \log (1 + x/4)}, & x \neq 0 \\ 12, & x = 0 \end{cases}$ is a continuous function,then the value of $k$ is

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 $[1, 2]$.