= Solution
{c}
For $0<\sigma^2<\infty$, the <central limit theorem> gives
$$
\sqrt n(\overline X-\mu)\xrightarrow dN(0,\sigma^2).
$$
If $g$ is differentiable at $\mu$, the <delta method> applies to the first-order expansion
$$
g(\overline X)-g(\mu)=g'(\mu)(\overline X-\mu)+o_p(n^{-1/2}),
$$
where the remainder notation means that its product with $\sqrt n$ tends to zero in <probability>. Hence
$$
\boxed{\sqrt n\{g(\overline X)-g(\mu)\}\xrightarrow dN\!\left(0,\sigma^2[g'(\mu)]^2\right),\qquad
g(\overline X)\mathrel{\dot\sim}N\!\left(g(\mu),\frac{\sigma^2[g'(\mu)]^2}{n}\right).}
$$
The useful nondegenerate <normal approximation> requires $g'(\mu)\ne0$. When $g'(\mu)=0$, the first-order limit is zero and a higher-order expansion may be needed.
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