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:
- 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.
- 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:
- 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.
- 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.
