Yeah, I meant luck in the sense that, if the author consciously used the first-order expansion over the true expression, the article doesn't state that only goi

Yeah, I meant luck in the sense that, if the author consciously used the first-order expansion over the true expression, the article doesn't state that only going to first order was reasonable. Hence one has to assume no such check was made. Just stating the calculation like he did might induce others to think that the expression applies regardless of the magnitude of the result. And then it will be pure luck whether it works or not.