Precalculus With Limits 4e Answers
Maya Malbon
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 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).
I just started studying calculus by myself, and I am in limits, but I don't seem to understand why the result changes when I use algebra. I understand how to do it. But why is the result different if it is supposed to be the same expression?as an example:
I guess that what we are doing is adding some kind of restriction, where $(x-2)\neq 0$. If that is true, then why we do not specify this in our result? (Or if we are meant to say it, I have not found such reference and I'll be glad if you can refer a source for this to me).
What bothers me is that I have not read any justification about when to use algebraic manipulations further than "to avoid indeterminates" or "to be able to find the limit" and so on. This helps with its objective, but does not explain why this is valid, or at least I cannot see it.
Once we've got it into the form of a continuous function (by which I roughly mean "you could draw it without taking your pencil off the paper", with some technical difficulties that mess that up) defined on a neighbourhood of the point (by which I mean you don't have any undefined-ness happening at the point, or at a sequence of points converging to it), you can make use of the fact that the limit of a continuous function at a point that it is defined in a neighbourhood of is equal to its value at that point: since $x^2 + 2x + 4$ is continuous and defined everywhere (unlike $\fracx^3 - 2^3x - 2$, which is not defined at $x = 2$), we can just substitute in $x = 2$ to find the limit.
If you're interested in going into more detail on this sort of thing and being more careful about these kind of issues, that's generally what is referred to in English as Analysis (as opposed to Calculus, which is the more handwavey type that you've seen so far). There are plenty of online resources for the subject - if you let us know what your first language is, I'm sure someone will be able to suggest resources in that language.
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