Asymptotic normality of $\bar X/S$ for exponential data, via the delta method

Suppose $X_1,X_2,\dots,X_n$ are independent and identically distributed random variables each having exponential distribution with mean $\theta$. Then the asymptotic distribution of $\sqrt n\left(\dfrac{\bar X}S-1\right)$, where $S^2=\dfrac1n\sum_1^nX_i^2-(\bar X)^2$, is: