Suppose one knows that for a random function $f(n)$, $f(n)-a$ decays at some rate given by: $$Pr(|f(n)-t|>\epsilon)=g(\epsilon),$$for $g(\epsilon)\to0$, all as $n\to\infty$.
If the above holds, then plainly in addition to $f(n)\to a$, we have $1/f(x)\to 1/a$.
But at what rate?
Since $$ \bigg| \frac1{f(n)}-\frac1a \bigg| = \frac1{|af(n)|} \cdot |f(n)-a|, $$ as soon as you know the order of magnitude of $|f(n)-a|$ and $|f(n)|$, you know the order of magnitude of $\big| \frac1{f(n)}-\frac1a \big|$.