Evil Limiting Function

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 $x$ , $\{x\} = x - \lfloor x\rfloor$ .

• $g\circ f(x) = g(f(x))$ .