Are convergent and divergent sequences a prerequesite for calculus or are they a part of it?

Question

Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: http://www.stitz-zeager.com/szprecalculus07042013.pdf

However, it does not cover convergence,limits etc. and those topics were briefly mentioned in my old textbooks. So what i am asking is are these topics a prerequesite for calculus or are they a part of the subject?

Answer 1

It depends on how Calculus is treated.

At my university and others I've attended (US), the concept of the limit is usually treated early in the first Calculus class in order to talk about continuity and derivatives. The core idea of sequences is brushed over in the introductory courses (except in the advanced courses).

Sometimes (and the department is debating this here) limits and sequences are taught in precalculus, allowing them to be reinforced in calculus. But in general, they're not expected to be well known before the first calculus course.

Personally, they're good to look into algebraically before calculus, but they involve topological ideas more suited to calculus.

Answer 2

Strictly speaking, the following might be slightly off topic, but I think that it is directly relevant.

From the preface of Calculus with Analytic Geometry by John H Staib [1966],

The development of the book is committed to the following thesis: The theory of limits is most easily grasped in the case of sequences. Therefore the theory of limits is presented first for sequences and then that theory is exploited in the introduction of all other limit concepts including integration. Thus, although most of the usual topics appear here, they are in a rather different order. Moreover, the emphasis on sequences not only provides the course with a unifying theme,but also give it a distinctive flavor which is reflected in many of the proofs and exercises. Indeed, it is the special allurement of sequences that I have attempted to exploit. For instance, few students are moved by the announcement that $\lim_{x\to 2}x^2=4$ but "all" students feel the challenge implicit in the assertion that the sequence $$\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, > \dots$$ has 2 as a limit

Answer 3

OK, here is a directly relevant answer: asymptotic expansions, which are finite expansions by definition, are used to approximate $f(x_{0}+h)$ and can be used with divergent series (That is what Poincaré invented them for) as well as convergent series.

The second part of my answer is that, at least in calculus, but in fact in precalculus, polynomial expansions (certainly finite and in fact, rather short) should be used to describe the qualitative behavior near $x_{0}$ and $\infty$ of the usual functions.

For a simple example, consider the function $f(x)=\frac{x^{3}-8}{x^{2}-5x+6}$.

1. Near $\infty$, we get by long division in descending powers that $f(x) = x +5 + 19x^{-1} + o[x^{-1}]$ i.e. an oblique asymptote.
2. Near $+2$, one of two possible poles, we get by long division in ascending powers that $f(+2+h) = \frac{+12h +6h^{2} +h^{3}}{-h+h^{2}} = -12 -18h -19h^{2} + o[h^{2}]$ i.e. that $+2$ is regular.
3. Near $+3$, the other possible pole, we get by short division that $f(+3+h) = \frac{+19 +27h +9h^{2} +h^{3}}{h+h^{2}} = \frac{+19+ o[1]}{h+ o[h]}=+19h^{-1} + o[h^{-1}]$ i.e. that $+3$ is indeed a pole.
4. A smooth interpolation of the local graphs near $\infty$ and $+3$---the local graph near $+2$ provides only confirmation---gives the essential global graph which in turn shows that $f$ has at least one maximum and one minimum.

In precalculus, rather than little ohs, I just use [...] to stand for "something too small to matter here."

See, for instance, my implementation of the above: Reasonable Algebraic Functions

Answer 4

Just to throw out a different answer: my immediate reaction was 'neither'. My experience in the UK has been that limits and convergence get the slightest whisper when you first meet differentiation and integration at high school, and at university they sit firmly under the heading 'analysis'.