Delta Method for Square Root of the Sample Mean

Let $X_1, \ldots, X_n$ be i.i.d.\ $\operatorname{Exp}(\lambda)$ with $\lambda = 4$ (so $E[X_i] = 1/4$, $\operatorname{Var}(X_i) = 1/16$). Define $T_n = \sqrt{\bar{X}_n}$.

(a) Using the delta method, find the asymptotic distribution of $\sqrt{n}\,(T_n - \sqrt{\mu})$ where $\mu = E[X_i]$.

(b) For $n = 256$, approximate $P(T_{256} > 0.525)$.

You may use $\Phi(1.60) \approx 0.9452$.