= Solution
Let $S_n=\sum_{i=1}^nX_i$. The <log-likelihood> is
$$
\ell_n(\theta)=n\log\theta-\theta S_n,
$$
with <score function> $n/\theta-S_n$ and second derivative $-n/\theta^2<0$. The unique <maximum-likelihood estimator> is therefore
$$
\boxed{\widehat\theta_{\rm MLE}=\frac n{S_n}=\frac1{\overline X}.}
$$
For an <exponential distribution> of rate $\theta$,
$$
\mathbb E[X_i]=\frac1\theta,
\qquad
\operatorname{var}(X_i)=\frac1{\theta^2}.
$$
The <central limit theorem> gives
$$
\sqrt n\left(\overline X-\frac1\theta\right)
\xrightarrow dN\left(0,\frac1{\theta^2}\right).
$$
Applying the <delta method> to $h(x)=1/x$, for which $h'(1/\theta)=-\theta^2$, yields the <exponential-rate maximum-likelihood estimator> limit
$$
\boxed{
\sqrt n(\widehat\theta_{\rm MLE}-\theta)
\xrightarrow dN(0,\theta^2).}
$$
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