Let $x>0, x \neq 2$, and let $C_{x}$ denote the positively oriented circle of radius $x$ centred at the origin. Define

$$g(x)=\oint_{C_{x}} \frac{z^{2}+e^{z}}{z^{2}(z-2)} d z$$

Evaluate $g(x)$ for $x \in(0, \infty) \backslash\{2\}$.