Precalc courses differ a lot, so it's hard to say something is included or not. After all there was a time when there didn't even exist a precalc course (different from a strong algebra 2, trig, and nalytic geometry sequence). Then again there are some precalc course that spend a reasonable amount of time on theory of functions (and relations), inequalities, and include limits and even simple differentiation, anti-derivatives. Basically an intro to calculus.
The sequence I had was:
*Pre-algebra (solving for single variable in first order equations)
*Algebra 1 (lotta x-y lines and maybe the quadratic equation)
*Geometry (lotta triangle proofs)
*Algebra 2 (logarithms, exponents, sequences, basically "college algebra", except a very short intro to vectors (needed to support HS physics
*Trig (1 sem)
*Functions (1 sem)
*AP Calculus (AB or BC)
The upper track kids did the sequence above in 7-12. Taking "freshman algebra" as eighth graders and algebra 2 and trig being combined from 3 to 2 semesters in a course designed for that track. The standard track started with algebra in year 9 (prealgebra in 8). Finished with either functions (2nd semester senior year) or with analytic geometry (kids taking algebra 2 trig or maybe doubling up a semester somewhere...I know "functions" was required for AP Chem so that provided a driver for some kids who were not in 8th grade algebra. I believe it was also acceptable to stop with geometry in year 10, for the lowest track.
The "functions" course corresponded to a strong pre-calc course as described above (but not some real analysis monstrosity). And also maybe just gave some time/maturity/practice that would be helpful before going to the shock of college calculus (BC is very comparable to a classic college calc class). Similarly for analytic geometry. I mean in theory, you could go straight from trig to a classic calc course, with analytic geometry in the text also. But I think you'd have a lot of attrition, given the sudden jump in difficulty.
As a single datum, the (strong) local public high school I went to in the 80s, did have limits in precalc. And at a reasonably rigorous level, more than just "derivatives are like tangents". Talking about left and right hand limits and where a limit doesn't exist. And the darned Le Hospital. But without such a concentration that people become more interested in the rigor as opposed to the result).