Suppose a sequence of random variables $X_n$ converges in probability to a random variable $X$. We know that $X \geq 1$. Does the following Taylor expansion hold? $$\frac{1}{X_n} - \frac{1}{X} = -\frac{1}{X}(X_n - X) + O_p(|X_n - X|^2)$$
If $X$ is a nonrandom number, then this is just the delta method. But now $X$ is a random variable, the delta method argument no longer applies.