Perhaps this is not a good answer per-se, insofar as it yet-once-again questions (to a certain degree) the premises of the question...
But, indeed, in many propitious situations, everything in sight is continuous, and it is justifiable to replace sub-expressions with their limits. I might claim that the practical success of calculus substantially depends exactly on the fact that "doing the obvious thing" very often succeeds in practice. Indeed!
In that context, it is mildly perverse (I do claim) to make beginners worry about bad possibilities that (with luck) do not occur at an entry level. Further, I would claim that the physical intuition that students have (naive as it is) is fairly correct about good limits.
I tell my first-year calculus students that if plugging in the limit value produces a sensible value, it is very likely that it is correct. And I also tell them that if their or other peoples' lives depended on it, they'd want more assurance than that.
So, seriously, what bad limits will people have reasonable motivation to worry about? I think the difficulty many beginners have in taking "the rules" seriously is that all they see are contrived examples quite-obviously designed to "punish" the non-conformist.
That is, the success of calculus and such is that mostly it does work as one might imagine, and even better. It was (as I've ranted many places) not a big hit because of invitations to fretting about rigor, but because it decisively answered questions about real things (whether in the external physical world or in the more-internal mathematical world).
And, for example, I cannot bring myself to advocate "conformance to rules", but I can easily advocate "viewpoint that explains things". Yes, it is not easy to find subtle mathematical issues that have much sense or moment for scientifically naive kids. I conclude that subtle analytic issues should be delayed until they might genuinely matter.