Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to know, why would the definition be difficult to understand? Too many logical quantifiers? The definition of "sequence" itself is vague? The usual examples are not enough to develop intuition?
I hope this question is not too vague.
On
I think my confusion of limits of sequences came from the definition of 'precision' that I was used to. You can say something is precise to mean that it is, in a sense, not messy. For example, 3 precisely divides 9, and we can write $a^2-b^2$ precisely as $(a-b)(a+b)$.
When we talk about dynamics or suchlike, quite often we use Taylor's theorem to get an approximation, and then things become imprecise. We started to simplify things, ignore forces, assume point masses, or spherically symmetric bodies, etc. This seems like a rather messy way to go about things (although this is not intended to spark an argument about whether or not this is true, it is just an opinion!)
So when we talk about limits, I suppose it helps to work with a different definition of precise (or at least, it helped me considerably).
When we talk about 'real-world' mathematics, nothing is exact. Circles don't exist, lines aren't one dimensional, and so on. But, you tell me exactly what you mean by 'precise', and I can give you an answer that satisfies your definition. It's not as if I can give you an estimate; I could literally go on forever refining my answer to whichever standards you set me. (That is, given any $\varepsilon>0$ I can find a term that is at most that far away from what I define as the limit, and I can carry on forever as $\varepsilon$ gets smaller).
I am not trying to explain what a limit is here, because you obviously already know, but what I hope to put across is how I had to change my idea of what was 'messy maths' and what was 'neat maths', and how limits seemed to make more sense when thinking of them as part of applied maths than pure.
Hope that helps in some way!
On
I believe the problem with introducing limits is that the quantifiers might be confusing for students looking at it for the first time. Instead of a symbolic definition, it is important to explain it pictorially, to begin with, and then move on to the "formal" version. I believe "seeing" something pictorially enables the students to understand the key ideas succinctly and conveys to us a depth of understanding that a symbolic description cannot easily match. John Morgan Allman, in his book "Evolving brains", claims that evolution has worked $1000$ times longer on our perceptual abilities than our symbolic-reasoning abilities. A simple example to start with would be say to consider a sequence
$$x_n = 1-\dfrac1{2^n}, \text{ where }n \in \mathbb{N}.$$
You can ask the students to mark these points on the "real" line
and hopefully they draw something like the above. They should now believe that the sequence is less than $1$ and can be made arbitrarily close to $1$. Once they have this picture in mind, you can have a formal definition of the limit of the above sequence.
You can then ask the students to consider different real sequences and ask them to mark the real sequences on the "real" line. For instance, a real sequence that converge to $1$ but oscillates around $1$. Once they have played around with marking these real sequences on the real line, you can then introduce the symbolic definition, with the different quantifiers, to define the limit of a general sequence.
On
This answer is a mere recall from my first exposure to calculus from a horrible, condescending instructor whom made calculus the most difficult subject for me. I am certain now though, that I am better than her.
That aside, the reason limits were so difficult are the following
(i) Calculating limits requires many basic foundations which most students lack, these includes absolute values, distributive properties of polynomials, knowing certain properties of functions (trig, exponential, rational functions, etc...). Most exercises involve combining everything they learned in pre-calculus which most lack
(ii). Other reason is because calculus is the first time they will have to look at math differently. I mean I wasn't very comfortable dealing with the infinite, it was fine and dandy counting finiteness, but infinity! It's also difficult to accept infinity as a mere concept rather than a thing, I blame this on the rubbish they broadcast on the science channel to fool the uneducated.
On
If you haven't done it already, please do look through Cauchy's rigorous definition of limits and continuity. If you are comfortable with the algebra of inequalities, the idea of limits becomes very much rigorous than intuitive. (Personally I found it a lot easier doing it the "Cauchy way")
See this. It discusses Cauchy's contribution to calculus and how he made calculus a rigorous mathematical field in a time where various (rather unknown) mathematicians logically (and beautifully) pointed various gaps in Newton's and Leibnitz's beautiful intuitive brain-child.
As far as I see it, there are two massive obstacles that a classroom must overcome when trying to come to grips with formal limits.
A big problem with understanding the limit definition, as you have pointed out, is the problem of quantifiers. There are so many, and the order in which they are employed is so important, that it can be difficult to keep track of. (And just when they think they understand, now they have to deal with uniform continuity, which to most will look exactly the same as continuity!)
To make matters worse, it's hard to describe a limit in a language that is precise enough to be useful to solve problems, but gentle enough to be swallowed easily. This is hardly surprising; literally thousands of years of mathematics was done before anyone really found the need to understand them completely.
A big problem with using the limit definition, is that it feels circular. This is often the first time students will have to consciously work ''forward'' to first convince themselves what the limit is before they can prove it.
For example, when you are solving $3x+4=5$ there is a mechanical list of procedures that one can go through to get to the answer. Even when doing something more technical, like verifying the trigonometric identity $\tan^2x + 1 = \sec^2x$, you can still start from what you're given and come up with the answer. Thoughtful, informed students will know this logic needs to be run in reverse to constitute a proof, but there is at least a foothold for one to grab onto; that original equation.
But when you are trying to find the limit of $x^2$ as $x\to 4$, if you don't have the intuition that the answer should be $16$, your work becomes very difficult. You can apply the limit definition to try to find a $\delta$, but the catch-22 is that you need to ``know'' what the limit is before you can even interpret which points are in the $\varepsilon$-neighborhood..