Suppose we define \(f(x) = \text{sgn}(\sin(x)) + \{ x\} \) for \(2\leq x\leq4\) and \(g(x) = -2 +|x-3| \), find the value of \( \displaystyle \lim_{x\to3} g\circ f(x) \).

**Notations**

\(\text{sgn}(x) \) denotes the signum function of \(x\), \( \text{sgn}(x) = \begin{cases} 1 \quad,\quad x>0 \\ 0 \quad,\quad x=0 \\ -1 \quad,\quad x<0 \end{cases} \).

\( \{ x\} \) denote the fractional part of \(x\), \( \{x\} = x - \lfloor x\rfloor \).

\( g\circ f(x) = g(f(x)) \).

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