# Teaching limits of sequences before limits of functions in Calculus?

Most Calculus courses/textbooks I have seen teach the different topics in that order: limits of functions, continuity, differentiability, integration. Then, depending on the teacher/textbook's preference, sequences, series, power series, differential equations, curves are taught in various orders.

I want to discuss the following variation: what are the pros and cons of teaching first limits of sequences (I assume that generalities about series - definitions $a_n=f(n)$ or $a_n=f(a_{n-1})$, increasing, decreasing sequences,... - is part of a preCalculus class but it could be included here) and then limits of functions, continuity, differentiability, integration,...?

Pros:

1. I think many students have difficulties to understand all the various $\epsilon-\delta$ definitions of limits of functions. For a function $f(x)$, you need to understand the different meanings of the locution "is close to" for the input $x$ (close to a point $a$, from the right $a^+$, close from the left, "close to $\infty$"="large enough" and "close to $-\infty$"="negatively large enough"), as well as for the output $f(x)$. I found that some students are confused by all these possibilities. For a sequence $\{a_n\}$, everything simplifies for the input since $n$ can only tend to $\infty$. Thus, studying sequences first allows to slow down the pace of learning $\epsilon-\delta$ definitions by considering only the output first.
2. I find that sequences and the Sequential Characterization of Limits (one of my favorite theorem :D) are very convenient to prove that a limit of a function does not exist. For example, showing that $\sin$ has no limit at $\infty$ is a piece of cake with the SCL. It can also be used for functions of several variables and I think it is simpler to write why a given function has no limit with the SCL than by considering limits along a curve.

Cons:

1. The only disadvantage I see with this order of topics is that you probably need to spend time reviewing sequences before starting series.

My question: are there any other advantages and disadvantages in starting a Calculus course by studying limits of series?

Subsidiary question: any reference for a textbook following this particular order?

• my colleague who wants to see real analysis taught before calculus would agree with this approach. Honestly, there are a few advantages to this. I wouldn't throw out the epsilon-delta, rather, teach this as another method. If you did this then the Riemann sum has a little less weirdness since you have discussed limits of sequences. Many presentations just throw $n \rightarrow \infty$ into the mix with integration without giving a formal definition of what is meant by such. The students don't notice because they're pretty lost in the sea of new ideas, but later... it will dawn on some the cheat Jul 13, 2014 at 5:18
• Using the sequential characterization also makes proving some of the limit properties far more approachable, in my opinion. I think that students are able to visualize a convergent sequence far more intuitively than a convergent function. Oct 27, 2015 at 12:16
• @JamesS.Cook If I were in that position, I would expend some time demonstrating that the $\epsilon - \delta$ and sequential pictures are equivalent, though I don't think real analysis should be taught before Calculus, it would be neat to see a program where secondary students on a math track took a co-requisite analysis elective. Oct 27, 2015 at 12:19
• Note for educators: When teaching calculus, get a good textbook and follow it closely. Someday, after you have lots of experience, when you write your own textbook, you may want to consider rearranging the order of the topics. Feb 19, 2016 at 23:58
• Two books that treat sequences before limits of a function of a real variable: Courant’s (with or without John) Calculus and Hardy’s A Course of Pure Mathematics Jul 24, 2018 at 1:39

To explain the answer, I found this question while thinking about asking a similar question. In teaching both pre-calculus and calculus, this is an issue that seems to come up over and over again in my professional life—students need to be given some notion of limits so that the rest of calculus can be made to work, but the full rigor of $\varepsilon$-$\delta$ arguments are not appropriate (here, I am thinking of a US-centric curriculum, with a student population that consists mostly of non-math majors who will never take analysis).

In the last year, I have been experimenting more with a sequential-first approach to teaching limits. In addition to your list of pros, I would suggest the following as motivation:

Pros:

1. I think that sequential limits are more intuitive. In fact, something that I see students do a lot on exams is try to compute limits sequentially. For example, they might try to compute $$\lim_{x\to 5} \frac{x^2-25}{x-5}$$ by evaluating the rational expression at several values of $x$ that are close to 5. They are clearly thinking of some kind of sequential definition of a limit, hence I think that it is reasonable to harness that intuition and make it rigorous.
2. In some sense, sequential limits are "simpler". When taking the limit of a function at a point, there are two quantities to keep track of: the limit point in the domain, $\color{red}x$, and the values in the codomain, $\color{blue}{f(x)}$: $$\lim_{\color{red}{x\to a}} \color{blue}{f(x)}.$$ In contrast, all sequential limits are taken at infinity. We don't have to worry about the points in the domain, and can focus on the behaviour of the function (i.e., the sequence) itself. I am almost tempted to introduction notation like $$\operatorname{Lim} a_n := \lim_{n\to \infty} a_n \qquad\text{or}\qquad \operatorname{Lim}\, \left(a_n\right) := \lim_{n\to \infty} a_n$$ to denote a sequential limit, thereby removing some extraneous notation from the first introduction to a limit (I've not tried this yet, and am still not convinced that it is a good idea, but the overall point is that we don't have to worry about notation for $n$ going to infinity—at worst, the notation $\lim_{n\to\infty}$ has some extra decoration which we can explain later).

There is, unfortunately, a fairly important downside to this approach that I think has been missed, which is going to take some explanation to get to. If we want to be really rigorous, we would probably start with a good definition of a sequential limit (your students might not see this, but it should be in your notes somewhere, I think):

Definition: Let $(a_n)_{n\in\mathbb{N}}$ be sequence and let $L \in \mathbb{R}$. We say that $$\lim_{n\to\infty} a_n = L$$ if for any $\varepsilon > 0$ there exists some $N$ sufficiently large such that $$|a_n - L| < \varepsilon$$ whenever $n \ge N$.

Because the Greek letter $\varepsilon$ is often kind of scary, I typically write $\text{error} > 0$ and $|a_n - L| < \text{error}$ when I first introduce the idea, then transition to Greek a bit later. I also try to spend a good amount of time explaining the basic idea of this definition, i.e. we set an error tolerance, then seek to find a value of $N$ that is so large that every term of the sequence is closer to the limit $L$ than the error tolerance.

The alternative idea is to pull out a calculator and work a bunch of examples. For example, we might consider $$\lim_{n\to \infty} \frac{ 5n^2 - 8}{3(n+1)(n+2)},$$ and try to get some intuition for the limit by evaluating this expression at various large values of $n$ (say, $n=1$, $10$, $100$, $1000$, $10^6$, and so on). It becomes apparent fairly quickly that this is converging. In an introductory class (i.e. not real analysis, not an honors calculus class; but maybe a regular calculus and definitely a pre-calculus class), the examples are generally enough to drive home the point, and a rigorous definition can be elided.

Once an intuition about sequential limits is developed, the next step is to work with continuous or functional limits. The "right" definition in this case is something like the following:

Definition: Let $f : X \to \mathbb{R}$ be a function defined on some set $X \subseteq \mathbb{R}$, let $L \in \mathbb{R}$, and assume that $I$ is an open interval such that $I\setminus\{a\} \subseteq \mathscr{D}(f)$. We say that $$\lim_{x\to a} f(x) = L$$ if for any sequence $(x_n)$ such that $\lim_{n\to\infty} x_n = a$ we have $\lim_{n\to\infty} f(x_n) = L$.

A lot of the technicalities here can be elided—the real issue here is the use of quantifiers. In order for the limit of $f$ to exist at $a$, we require that for every sequence $(x_n)$ such that $x_n \to a$, we have $f(x_n) \to L$. You can't just check a few examples—you have to check every possible sequence, which is hard. Counterexmples are easier to deal with, since it is necessary only to find one sequence which fails to converge (e.g. consider $\lim_{x\to 0} \sin(\frac{1}{x})$), but actually showing that anything converges is much harder. In short:

Cons:

1. Students at an introductory level have great difficulty with quantifiers. The fact that $\lim_{n\to \infty} f(x_n)$ must be $L$ for all possible sequences $(x_n)$ which tend to $a$ is a really, really difficult idea for many students.
2. Beyond just coming to terms with quantifiers, showing that something is true for every sequence $x_n \to a$ is a pretty big leap for most students.

In an introductory class, we can elide many of these these issues, and claim that it is "good enough" to check several sequences in a couple of key examples, then move on to continuity (and use properties of continuity to sort of "backfill" the skipped technicalities).

In a class for math majors, this becomes much more delicate, and I think that there is no way of avoiding the eventual introduction of the $\varepsilon$-$\delta$ definition (though I don't think that the original question is seeking to avoid such definitions; only to put them off a bit).

A reference:

Regarding references, I don't really have any that are really appropriate for an introductory class. However, a text that approaches real analysis from a chronological or historical approach (rather than an axiomatic approach) may give some good ideas. For example, I recently picked up a cheap used copy of Saul Stahl's Real Analysis: A Historical Approach, which seems (so far) to be a pretty solid text.

• +1. Simple things first. Sequences -> continuity -> limits -> derivatives... Jul 24, 2018 at 14:36
• If I were to introduce alternate notation, I think I would use $a_\infty$. It's convenient and compact notation that is actually used by mathematicians, and it might help drive home the point that students often miss that the limit is just a number, not some ephemeral, varying thing (and related misconceptions about $\infty$). It also seed the ideas of continuous extension and of extended real numbers. The main drawback I can see is that teachers in future classes might be uncomfortable with the notation.
– user797
Jul 25, 2018 at 7:42

According to me, another disadvantage of this approach is that limits of functions have a visual aspect because of the graphic representation of functions which can help to understand concepts.

When you look at the formal definitions of several concepts in real analysis you see that those concepts are rooted in the limit of sequences (or better to say, these concepts can be defined in a way that they all base on the concept of the limit for sequences):

• The limit of a series $\sum_{k=0}^\infty a_k$ is defined to be the limit of the sequence of the partial sums: $\sum_{k=0}^\infty a_k := \lim_{n\to\infty} \sum_{k=0}^n a_k$
• We have $\lim_{x\to a} f(x)=c$, iff for each sequence $(x_n)$ with $\lim_{n\to\infty} x_n = a$ we find $\lim_{n\to\infty} f(x_n) = c$.
• After we have defined the expression $\lim_{x\to a} f(x)=c$ with the limit of sequences we can define the derivative via $\lim_{h\to0} \tfrac 1h [f(x+h)-f(x)]$.

Thus, when you want to teach calculus hierarchically in a way that you only define new concepts with already defined terms, you need to start with the limit of sequences.

However, when we already know important concepts like the derivate or the integral, the need of talking about the limit of sequences can be easily motivated: We need this concept to define the other concepts rigorously. This motivation can be hardly given when we begin calculus with the limit of sequences. Note also that in mathematics a concept was often first used intuitively before the formal definition of this concept was established.

• The limit of a sequence is a special case of the limit of a function, since a sequence is just a function from $\mathbb{N}$ to some set (usually $\mathbb{R}$). So I find it hard to argue that the limit of a sequence is more fundamental than the the limit of another type of function, say $f:\mathbb{R} \to \mathbb{R}$ at some point $a\in \mathbb{R}$. The definition of limit is also not simpler if we restrict to sequences. Mar 2, 2020 at 15:11
• @MichaelBächtold: It depends which definition of the limit of function you mean. The limit of a sequence is of the form $\lim_{n\to{\color{Green}\infty}}$ while the limit of a function at some point is of the form $\lim_{x\to{\color{Green}a}}$. However you are right: Both definitions have the same structure (when you use the epsilon-delta definition for the limit of functions), which is the limit definition for topological spaces: math.stackexchange.com/questions/835978/… Mar 5, 2020 at 12:37
1. you don't need epsilon delta to cover limits of functions. Just derivatives.

2. (As someone else said) you lose the visual aspect of an asymptote.

3. You have more messy algebra with the terms of the series.

Other aspects of series/sequences taught later in the course (e.g. Maclaurin and Taylor). And if you move it all to earlier, it just delays learning some calculus (min/max problems, rates, etc.)

Net, net: There is a good pedagical reasons for the current order. This is another suggestion from someone who thinks in terms of already knowing the material and thinks in terms of logical exposition versus the most incremental steps for a new learner.

• What does "you don't need epsilon delta to cover limits of functions. Just derivatives" mean? How do you calculate $\lim_{x \to 0}|x|$ using derivatives? How do you prove the nonexistence of $\lim_{x \to 0}\tfrac{|x|}{x}$ using derivatives? Jul 24, 2018 at 8:24
• "Net, net: There is a good pedagical reasons for the current order." Can you expand on this? That's the core of the question so it would improve your answer if you could detail this point. Also, snarky comments like "This is another suggestion from someone who thinks in terms of already knowing the material" don't help the answer. Jul 24, 2018 at 15:12