About the word "limit" used in calculus

Question

In the introduction of the limits chapter of a scholar book I can read (translated and abstract) the following examples:

1. "the price of a product has a limit value that, from this price, the number of sales start to decrease, reducing the benefits"
2. "the light speed has a limit"
3. "the population of animals is limited due to the amount of food"

For me, these examples are not correct examples of a limit in the sense used in calculus: 1 and 3 are a function maximum, example 1 is a maximum of the function f:price->benefit and example 2 is a empirical maximum, base of the special relativity theory (I do not known any formula that gives light speed as root, maximum or limit of an expression ).

It is true that in a lot of situations in normal life, limit and maximum are synonyms. But not in math.

Thus, I wonder if in the classroom we must strengthen the usage of words as "tendency", "trend", ... even when in the written math expression still using $\lim$.

Answer 1

One of the difficult aspect of learning and teaching mathematics is that it is in part akin to learning or teaching a new language. Words are used with precise technical meanings that are often divorced from their colloquial, everday uses. A simple, commonly problematic example is the word "sphere", that a mathematician uses to mean the surface of a perfectly round object, but which is used by many as a synonym for the solid that the mathematician calls a "ball". An example that vexes students of linear algebra is the use of "vector" to refer to an element of an abstract vector space, for example a space of polynomials or matrices.

My sense is that this issue is best addressed openly and explicitly, by explaining to students that the mathematical use of whatever terminology is in play, although perhaps partly motivated by some (perhaps long ago) colloquial use, is different from the colloquial use, and has a particular meaning specific to the context of its mathematical use. So, in a mathematical context, a mathematician would speak of a maximum admissible speed, or an absolute bound on the speed of a moving object, reserving the word "limit" for reference to a particular mathematical construct.

At early stages of learning this problem is sometimes even more pronounced because the distinction between mathematical speech and colloquial speech is emphasized less, or sometimes intentionally set aside. When an algebraic geometer speaks of "perverse sheaves" there is not much danger of thinking degenerate thoughts about agricultural products, but when a function is defined to be convex if its graph lies below its secant lines and the student is also learning about convex and concave lenses in physics, there is a lot of potential for confusion, particularly if no one points out to the student that the words convex and concave as used technically by mathematicians and physicists are not exact synonyms, at least not without some additional clarification.

Answer 2

This is an answer about terminology in general. (As always, I'm answering as a learner who has thought about the learning process rather than as a tescher, but I hope it will be relevant.)

I think that in order to learn the true meaning of a term, one must first understand the concept that it refers to. Ideally in that order: if possible, encounter the concept first and then learn the name for it. (This is the same as when learning a foreign language.) Then when you do learn the name—probably immediately after learning the concept—you're not misled by the common meaning of the word. You already know the meaning: what you didn't know was the word for it.

For example, I think I was taught about limits not long after getting used to the idea of asymptotes in the context of curve sketching. First we were shown some examples of sequences and sums like

$$S_n=1+\frac12+\frac14+. . .+\frac{1}{2^n}$$

We spent a bit of time playing with these and learning how to express the sums in shorthand using the $\Sigma$ sign

For the above example, it was easy to see that increasing $n$ made $S_n$ get closer and closer to $2$ without ever quite reaching it. And once we'd understood that, we were ready to be told something like

$2$ is called the limit of the sequence as $n$ tends to infinity and we write it as $$\lim_{n\rightarrow \infty }S_n =2$$

At this point there was nothing confusing about the word limit because we were already familiar with what it was describing. Tends to seemed a slightly odd phrase, but was just an instance of mathematics sometimes using funny language.

The same went for the notation: we weren't introduced to a strange new symbol followed by a definition to grapple with and misunderstand—we were introduced to something needing the symbol, then to the symbol itself.

Summary: I don't think terms which differ from their ordinary usage have to be confusing, provided they're not the first thing we encounter. After you've seen what the limit of a series is, the most obvious English word for it is limit, so it's not a problem being asked to use that word. The problem arises if we try to get the concept from its name, rather than the other way round.

Answer 3

These are limits

1. "the price of a product has a limit value that, from this price, the number of sales start to decrease, reducing the benefits"
2. "the light speed has a limit"
3. "the population of animals is limited due to the amount of food"

Each of these items above does, in fact, display limit behavior.

The economics of pricing and supply/demand is replete with curves and optimization that display limit behavior. Light speed exhibits limit behavior with time. As you asymptotically approach the speed of light, the effects of time dilation increase without bound. Animal population graphs are also asymptotic. As you add animals to an ecosystem, each of them is able to eat less and becomes slightly more likely to die of malnutrition or conflict. As time T goes to infinity, the animal population tends to an equilibrium expressed by the point at which the death rate of animals (due both to natural causes as well as causes related to food shortages) exceeds the maximum sustainable birth rate. This is the population limit.

To be clear, the first and third cases are not fully continuous (e.g. you can't increase the price of a good by 1/100,000,000th of a cent or increase the population of deer by 0.0000001 deer) and thus cannot be completely modeled by algebraic functions, but they can (and often are) modeled by curve-matching functions that do demonstrate limit behavior. The second is also likely not purely continuous due to quantum behavior, but it will be much closer and so your model will be much more accurate.