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$$\lim_{x\rightarrow a} f(x)=L$$

Which way should students best get in the habit of?

  1. The limit of $f(x)$, as $x$ approaches $a$, equals $L$
  2. The limit of $f(x)$ equals $L$, as $x$ approaches $a$
  3. The limit, as $x$ approaches $a$, of $f(x)$ equals $L$
  4. other (please provide)
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    $\begingroup$ I think I say it the 3rd way (which also keeps f(x) and L together). $\endgroup$ – Sue VanHattum Sep 3 at 3:05
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    $\begingroup$ The expression $\lim_{x\to a} f(x)$ has a meaning without the $=L$ part. Variant 2 makes this least explicit. $\endgroup$ – Michael Bächtold Sep 3 at 9:12
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    $\begingroup$ I think the best habit would be for the students to think they all mean the same thing. $\endgroup$ – user1527 Sep 4 at 13:21
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    $\begingroup$ A fourth possibility is "As x approaches a, the limit of f(x) equals L." Edit: Personally, I don't think I say it that way. But when I introduce the topic of limits, I might say things like "If x approaches a, what does f(x) do?" $\endgroup$ – idmercer Sep 4 at 16:01
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According to UCAR-10101: Handbook for Spoken Mathematics, Lawrence A. Chang, Ph.D., page 38, a source for how to speak mathematics to sight-impaired students, we have:

$$\lim_{x\to a} y = b$$ is spoken as the "limit as $x$ approaches $a$ of $y$ equals $b$". For the given expression, $$\lim_{x\to a} f(x) = L$$ is spoken thusly: the "limit as $x$ approaches $a$ of $f$ of $x$ equals $L$."

This is consistent with option $(3)$.


Alternatively, in analysis, from ANALYSIS TAUGHT BY BJORN POONEN, NOTES BY SANATH DEVALAPURKAR, we have the following:

We write $$\lim_{x→a} f(x) =L$$ to say that for every $\epsilon >0,$ $f$ is eventually within $\epsilon$ of $L$ as $x$ approaches $a$”. This means that there exists $δ >0$ such that $0<|x−a|< δ,$ then $|f(x)−L|< \epsilon$.


ADDED: Please see this mathematics.se post about "speaking mathematics". There are additional resources for those interested.

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    $\begingroup$ Thank you for the first reference---it is fantastic. I had never considered that someone might have codified spoken mathematics, and am quite chuffed to know that it exists. $\endgroup$ – Xander Henderson Sep 5 at 16:03
  • $\begingroup$ I am not sure what is the purpose of the three pages dedicated to Roman letters, as they offer no pronunciation. Decimal logarithm lg() is missing. He should have used non-square matrices to avoid ambiguity. Still, a nice find. $\endgroup$ – Rusty Core Sep 6 at 20:41
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    $\begingroup$ @Xander Henderson: I had never considered that someone might have codified spoken mathematics --- I hadn't either until a few years ago when I was involved with a few others in trying to implement JAWS into a certain high stakes test. I was quite amazed at the audio speed settings that competent visually impaired users apparently could manage. We made various spot-checks on the accuracy of the (continued) $\endgroup$ – Dave L Renfro Sep 7 at 17:09
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    $\begingroup$ formula rendering. Things like $3(a + \frac{b}{c})$ were easy ("three left paren frac b over c right paren" I think), but some of the items involved nested fractional expressions and various other complicated expressions, but the real problems were with graphs ($xy$ coordinate graphs, bar graphs, circle graphs, etc.), and especially tables of data. And then there's the problem of trying to equate scores for someone working under these conditions with sighted test takers, something that I think was mostly left up to graduate admissions departments. $\endgroup$ – Dave L Renfro Sep 7 at 17:14
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I say it the third way, for these reasons:

Firstly, from a notation point of view, the “$x\to a$” has to be written with the “$\lim$”, and no “$\lim$” can be written without it (without specifically saying what you mean by not having it), so it makes sense to put them together when you say it aloud.

Indeed, you could argue that $\displaystyle\lim_{x \to a}$ is an operation you perform on the function $f(x).$ That is, you can change what $x$ the limit is being found at and also what function you are doing it to.

Secondly the “$\lim_{x \to a} f(x)$” is a thing all by itself, and so it makes sense to say it all together.

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    $\begingroup$ from a notation point of view, the* “$x\to a$” *has to be written with the “lim”, and no “lim” can be written without it --- In mathematics one needs to be careful about strongly worded statements such as this. I've seen many instances in which a fixed point of application is used throughout (indeed, I was a referee for such a paper a few years ago) and hence excluded from the notation, and there are several cases of generalized/axiomatized notions of "limit" in which "lim" alone is often used to denote one of them (e.g. Banach limits, limit functors in category theory, etc.). $\endgroup$ – Dave L Renfro Sep 4 at 8:13
  • $\begingroup$ @DaveLRenfro I’ve made an edit. What do you think? $\endgroup$ – DavidButlerUofA Sep 4 at 9:16
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    $\begingroup$ I'm inclined to argue that since $x \rightarrow a$ modifies the limit operation, it is probably best not to separate "The limit" and "$x \rightarrow a$" with something else (e.g. "of $f(x)").$ I probably wouldn't side-track into whether we need to say $x \rightarrow a,$ as this is clearly context dependent, and in the present context it's clear we want to include it. But I wouldn't try to restrict myself to a single formulation (except in the early stages of a formal discussion of limits) --- see @Gerald Edgar's comment to another answer for why it can help to not have such a restriction. $\endgroup$ – Dave L Renfro Sep 4 at 12:44
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    $\begingroup$ +1. A few members of English Language Learners gave variations on this advice, to explain "How do you read these mathematical expressions aloud?" $\endgroup$ – Jasper Sep 4 at 22:46
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As $x$ approaches $a$, $f(x)$ approaches $L$.

First, we emphasize what is happening to the independent variable, then we explain the consequence. I think that this phrasing is concise and easy to understand. It is clean and efficient. This is essentially (3), but I think that the sub-clause "The limit..." is unnecessary.

Moreover, if we have a fixed function and want to consider limits at several points, it provides a consistent framework. For example, consider the rational function $$ f : \mathbb{R}\setminus\{\pm 1\} \to \mathbb{R} : x \mapsto \frac{x+1}{x^2 - 1}. $$ As $x$ approaches $\pm \infty$, $f(x)$ approaches $0$. On the other hand, as $x$ approaches $-1$, $f(x)$ approaches $1/2$, and as $x$ approaches $1$, $f(x)$ is unbounded (in either the positive or negative direction, depending on whether the limit is taken from the left or the right).

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    $\begingroup$ I've seen this written as $f(x) \to L$ when $x\to a$. But this need not be equivalent to $\lim_{x\to a }f(x) =L$. For instance: as $x$ approaches $1$, $x^2$ approaches $x$. But I cannot write $\lim_{x\to1}x^2=x$. $\endgroup$ – Michael Bächtold Sep 4 at 11:09
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    $\begingroup$ An advantage of this: You can say "as $x$ approaches $a$" once, then state multiple consequences of that without repeating it for each one. $\endgroup$ – Gerald Edgar Sep 4 at 11:14
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    $\begingroup$ @MichaelBächtold I prefer to keep my spoken mathematics a little looser than my written mathematics. Spoken language should get the ideas across in broad strokes. When precision is needed, we have notation which nails down our meaning. Hence I do not claim that my phrasing is exactly equivalent to $\lim_{x\to a} f(x) = L$. However, if $f$ is a function, $x$ is a variable, and $L$ and $a$ are numbers, then my phrasing is entirely unambiguous. Hence when speaking my mathematics, this is how I would phrase it. $\endgroup$ – Xander Henderson Sep 4 at 14:03
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I usually say

f(x) lähestyy L:ää, kun x lähestyy a:ta

and sometimes I instead say it more colloquially as

f(x):n raja on L, kun x on a.

The inverted

Kun x lähestyy a:ta, f(x) lähenee L:ää

is also fine and in use.

These would correspond to 2) and 4) in English. I would avoid linguistic complexity, such as a side clause embedded in the sentence, since it might reduce clarity.

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2
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You are interchangeable, like peas in a pod.

Cauchy and Weierstrass were usually saying "$f(x)$ becomes arbitrarily close to $L$", with the qualifier "as $x$ approaches $a$", sometimes before, sometimes after, sometimes implied.

They also followed Leibnitz and Lagrange to talked about a quantity $f(x)$ becoming infinely close to $L$, when $|x-a|$ is infinitely small or an infinitesimal.

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1 > 3 > 2

I prefer the first wording. You get more of the idea of thinking about what x causes what y. Yes, the second is equivalent, but it is awkward, to add the condition (approaching a) after the result. Three is OK also. Although I mildly prefer 1. Maybe shorter.

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    $\begingroup$ This seems more like a comment than an answer to the question, mostly because you only say things like "I prefer..." when the main question is about which one a student should use. $\endgroup$ – Brendan W. Sullivan Sep 4 at 15:41
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    $\begingroup$ Title has different wording. At the end of the day, it's not a super important question and difficult to parse too much of a long answer other than polling the forum. $\endgroup$ – guest Sep 4 at 23:35
  • $\begingroup$ Fair enough, but that just indicates this question should be improved, because conducting a poll to see what users say personally could be quite different from soliciting the "best one for a student to get in the habit of using". $\endgroup$ – Brendan W. Sullivan Sep 6 at 3:36

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