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 $\lim$.

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    I would continue to use the word limit in math class normally. 99% approaching asymptotically (haha) of these suggestions for changes in the traditional terminology and coverage are bad ideas. And they generally do NOT represent observed difficulties by the students but instead fussy observations of the instructor (often a TA or newish professor). – guest Nov 2 at 13:11
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    I agree with part of @guest's comment; I don't think you can change the terminology "limit". But I don't think your concerns are merely fussy. Students (in the U.S.) are certainly familiar with the phrase "speed limit" and probably with phrases like "limit of his endurance" or "limit of his patience", in which "limit" means maximum. So this dissonance between the mathematical meaning and other meanings of "limit" deserves to be discussed in class. – Andreas Blass Nov 2 at 13:25
  • @guest: thanks for your comment. 3 of 3 wrong examples in a book (if my rationale is correct) points to something deeper than a fussy observation. – pasaba por aqui Nov 2 at 13:37
  • @AndreasBlass: in fact, when we read "f(x)->4 when x->5" we say "tiende a" ("trends to" in english ?) not "limita con" ("limits with") – pasaba por aqui Nov 2 at 13:42
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    It's also correct in English to say "f(x) tends to 4 when x tends to 5" but it's more common to say "approaches" rather than "tends to". The real difficulty arises when you need a noun, as in "the limit of f(x) as x approaches 5"; I don't think there's any standard substitute for "limit" in this context. – Andreas Blass Nov 2 at 13:45

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.

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