I don't think the STEP markers will be particularly concerned about rigorous handling of limits etc.
There are a couple of things that come to mind but I suspect you already know them.
You're expected to be clear about how logical arguments work: the difference between "implies", "is implied by", "implies and is implied by". Things like proof using the contrapositive, what a counter example is, etc.
When the STEP examiners say things like "hence" or "use this result to...", you really have to use the previous result or you'll be docked marks. To be honest I think they're pretty harsh on this one; I generally think I'm pretty good at understanding "this is what they want you to do" but even so there have been several questions where I'd probably have lost marks for not doing things the way they expected.
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One that I don't think the examiners will mark you down for, but falls into "really annoy someone" (if that someone is me!): people writing down a load of lines of maths without anything connecting one line to the next. Actually having a logical connected argument will put you well ahead of the pack.