Mathematics Question from JEE Main 2023 · 30 January, Shift 2

Let $f$, $g$ and $h$ be the real valued functions defined on $\mathbb{R}$ as

$$f(x) = \left\lbrace \begin{array}{ll} \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{array} \right.$$ $$g(x) = \left\lbrace \begin{array}{ll} \frac{\sin(x+1)}{x+1}, & x \neq -1 \\ 1, & x = -1 \end{array} \right.$$

and $h(x) = 2[x] - f(x)$, where $[x]$ is the greatest integer $\leq x$. Then the value of

$$\lim_{x \to 1} g\big(h(x-1)\big)$$

is: