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I am cross-posting from MSE.

The students in my calc classes tend to be primarily bio/chem majors, and not very much math / physics / engineering.

I feel like there are pretty good ways to talk about and explaining the epsilon-delta definition of limits for physics, engineering students.

But I'm not exactly sure what could be a good one for chem / bio majors in a first year calculus class. A few examples were given in the comments in the MSE post, but I was hoping this could be expanded further.

One idea I had was something along the following lines:

Let $f$ represent the function that represents the blood sugar level as a function $x$ the amount of sugar (g) consumed per day. One wishes to maintain a certain a sugar level $L$, which can be achieved by consuming $a$ grams of sugar. But since it is difficult to maintain such a strict diet, one would like to allow for a bit of wiggle room without jeopardizing the sugar level too much...

Since I know very little about biology (or chemistry), I feel like this example may be too contrived or needlessly convoluted.

Any feedback or suggestion would be greatly appreciated.

Note: One big problem I have with many of the motivating examples from out side of mathematics is that they limit and continuity seem to be mixed up. Distinguishing the idea of limit and continuity already confuses a lot of first year calculus students even without the epsilon and delta thrown in there.

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    $\begingroup$ Personally, I would avoid epsilon/delta issues altogether for chemistry and biology majors, although I realize you may not have a choice in the matter. Instead (again, assuming you have a choice), put additional focus on rates of growth issues, especially why exponential functions model so many growth situations (the change in a quantity being proportional the the amount of the quantity is very prevalent in chemistry and biology). $\endgroup$ Commented Feb 3, 2015 at 22:01
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    $\begingroup$ Surveying the answers to a similar question, I don't see much motivation or justification for teaching epsilons and deltas. If most bio and chem majors won't appreciate proofs, I'd rather focus on other topics. See matheducators.stackexchange.com/questions/2060/… $\endgroup$
    – user173
    Commented Feb 4, 2015 at 1:00
  • $\begingroup$ First of all, there is a huge difference between motivating the concept of a limit, and motiving the formal definition of a limit. The former depends on why you're introducing limits to biology students in the first place (do they need to calculate derivatives or something?), whereas the latter is best explained as part of a general explanation of why mathematical precision is necessary. $\endgroup$
    – Jack M
    Commented Feb 6, 2015 at 20:21
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    $\begingroup$ Upon re-reading your post, I feel you're misusing the term "motivation" (although the meaning of the term is certainly somewhat subjective). A motivation for a definition is usually a reason why you would need to define such a thing in the first place, what you're looking for seems more to be a way of explaining why $\epsilon-\delta$ is a good definition for the intuitive concept of a limit, which I assume they already have. $\endgroup$
    – Jack M
    Commented Feb 6, 2015 at 23:40
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    $\begingroup$ @Braindead I think your explanation needs to take into account what they use limits for. We use one definition of limit for many different purposes in mathematics, but remember that it's not initially obvious that all of those different situations are well-modelled by a single common definition. Explain the particular application(s) that they're used to. If you explain it in terms of anything else, they'll be left wondering what that has to do with the math that they have to do. $\endgroup$
    – Jack M
    Commented Feb 7, 2015 at 20:55

4 Answers 4

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If you must introduce epsilon/delta...

As the sciences usually deal with real measurements with errors, this might be a way in: Is the calculation increasingly stable under decreasing measurement errors?

It is possible to motivate much of differentiation just from estimating errors. The percentage error (error / value) of squares is doubled, and of cubes is tripled. This is the definition polynomial differentiation.

This is the link Feynman made use of in his famous abacus vs mental calculation anecdote. As a bonus, an interesting anecdote that shows something is not as difficult as it first appears is always good motivation!

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(I should preface that I also agree with Dave Renfro (in the comments to the original question) that, given a choice, there is much more useful and relevant material to cover than the $\epsilon$-$\delta$ definition in a first calculus course for chemistry and biology majors.)

I think the problem you suggest is a good motivational problem, but that it is not strictly speaking motivating the $\epsilon$-$\delta$ definition. Your problem amounts to given $L$ and a specific $\epsilon > 0$, find a specific value of $\delta$ such that for all sugar consumption amounts $x$ that are within $\delta$ of $a$, the associated blood sugar level $f(x)$ will fall within $\epsilon$ of $L$. (FWIW, I don't know if the correlation between sugar consumption and blood sugar level is as simple as your function model assumes, though I think it sounds like a plausible first model to try. But you might try asking biologists or medical experts whether this model is reasonably accurate, an acceptable white lie, or an outright gross oversimplification of the situation.)

There are two key respects in which your example does not fully capture the definition of a limit:

(1) In practice, you would likely only be interested in a single $\epsilon$ value or perhaps a small finite number of $\epsilon$ values (different patients may require different levels of care for their blood sugar levels). It is important to point out that the mathematical definition requires the existence of a $\delta$ for every $\epsilon$ (and that you are allowed to use a different $\delta$ for each $\epsilon$). You can certainly use your example to help explain this, but it's important to recognize the quantification logic behind the definition of a limit gives students a lot of difficulty and consequently that it needs to be addressed carefully.

(2) Your example uses a continuous function, where $L$ will be equal to $f(a)$. The primary motivation for introducing limits is to deal with discontinuous functions, such as the average rate of change functions that arise in computing derivatives. It is precisely because the function which computes the average rate of change of $y = f(x)$ with respect to $x$ for $x$ in the interval $a \leq x \leq a + h$ (viewed as a function of the variable $h$) is not defined (and consequently not continuous) when $h = 0$ that limits are needed. This point does not invalidate your example, but you need to address this point. Inevitably, one must introduce an example involving a rate of change to fully motivate introducing limits.

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Just as you crossposted from MSE, let me crosspost heropup's tremendous answer, which I improved.


Imagine the following Socratic dialogue.

Teacher: What does $\lim\limits_{x\to a} f(x) = L$ mean?

Student: It means that the limit of the function $f(x)$, as $x$ approaches $a$, equals $L$.

Teacher: Yes, but what does that actually MEAN? What are we saying about the behavior of $f$?

Student: [Pauses to think.] I guess what we are saying is that for values of $x$ "close to" $a$, the function $f(x)$ becomes "close to" $L$.

Teacher: OK. So how are you defining the concept of "close to?" In particular, how does math quantify the notion of "closeness"? Does "close to" mean $x = a$?

Student: No — well — maybe sometimes! Of course, if $f(a)$ is well-defined, then we just have $f(a) = L$ — but this is plain vanilla, and trivial. The whole point of limits is to describe the function's behavior around the point $x = a$, even when $f$ ISN'T defined at $a$.

Teacher: Right, but you didn't answer my question. So how would you mathematically define "closeness"?

Student: [Long pause.] I'm not sure. Well, hold on. Let me try a geometric argument. When a number $x$ is "close to" another number $a$, we are really talking about the distance between these numbers being small. Like $2.00001$ is "close to" $2$, because the difference is $0.00001$.

Teacher: But that difference — which you call "distance" — isn't necessarily "small" in and of itself, is it? After all, isn't $10^{-10^{100}}$ much smaller than $10^{-5}$? "Small" is relative.

Student: [With irritation] Yeah, but you know what I mean! If the difference is small enough, then the limit exists!

Teacher: [Chuckles] Yes, I see what you're getting at! But so far, all you've been doing is choosing different vocabulary to describe the same concept. What is "distance"? What is "small enough?" We are mathematicians — how can improve this inaccurate and imprecise words? Take your time to think about this.

Student: [Sighs] What I was doing before, I was calculating a difference between $x$ and $a$, and calling it "small" if it looked like a small number. But what matters ISN'T the signed difference, but the absolute difference $|x - a|$. Since (as you put it) "small is relative," let's instead use a variable, say $δ$ (to abbreviate "difference"), to represent some bound... [trails off]

Teacher: Go on...

Student: All right. So if $|x-a| < δ$, then $x$ is "close to" $a$. We choose some number $δ$, in some way that quantifies the extent of closeness.

Teacher: OK. Is $δ$ allowed to be zero?

Student: Oh, of course not, no! I forgot. No, we need $\color{red}{0 < |x - a| < δ}$. Then $x$ is "delta-close" to $a$, or in a "delta-neighborhood" of $a$.

Teacher: All right. Now how are you going to tie $δ$ to the behavior of $f$?

Student: [Exasperated] Yes, yes, I'm getting to that part. As I said, the limit is something where if $x$ is "close to" to $a$, then $f(x)$ is "close to" $L$. Obviously, $f(x)$ can have a different extent of "closeness" to $L$, as $x$ does to $a$.

For example, if $f(x) = 2x$, then when $x$ is within $δ$ units of (for example) $1$, then $f(x)$ is only bounded within $2δ$ units of $2$, since $0 < |x-1| < δ$ implies that $0 < |2x - 2| = |f(x) - 2| < 2δ$. But functions can be arbitrarily (although not infinitely) steep. How can we quantify the relationship between the closeness of $x$ to $a$, as this closeness impacts the closeness of $f(x)$ to $L$?

Teacher: You actually touched on it, when you said that functions can be arbitrarily but not infinitely steep. Stated informally another way, it means that the function's value can change very rapidly — in fact, as rapidly as you please — but only finitely so, for some fixed change in $x$. So if you wanted to guarantee shrinking the difference between $f(x)$ and $L$ as small as you please, while not necessarily zero, how would you do it?

Student: [Long pause.] I need help.

Teacher: So far, you've been thinking about using (as you put it) "delta-closeness" to force $f(x)$ to be "close to" $L$. But what if you turned it around and instead said, I'll force $f(x)$ to be as close as I please to $L$? Then what does this closeness of $f(x)$ say about how close $x$ is to $a$? That way, you are guaranteeing that $f(x)$ becomes close to $L$, but the cost of that guarantee is that we need to guarantee that...

Student: [Interrupts] Oh, oh! I get it now! Yes. What we need to say is that for a given amount of "closeness" of $f(x)$ to $L$, a $δ$-neighborhood around $a$ where (if you pick any $x$ in that neighborhood) will guarantee $f(x)$ to be "close enough" to $L$ — that $f(x)$ will be within that given amount of closeness. In other words, we pick some "tolerance" or error bound between $f(x)$ and the limit $L$ that is our criterion for "close enough." And for that closeness, some set of corresponding $x$-values close to $a$ will guarantee that $f(x)$ meets the closeness criterion.

Teacher: Good, good. But how do we formalize this?

Student: Well, we need another variable to describe the extent of closeness between $f(x)$ and $L$...let's use $ε$, to abbreviate "error." As we did before, we use the absolute difference $|f(x) - L|$ to describe the "distance" between $f(x)$ and $L$. So our criterion has to be $\color{lightseagreen}{|f(x) - L| < ε}$. This time, we get to pick $ε$ freely, because it represents how much error we will tolerate between the function's value and its limit. We must be able to choose this tolerance to be arbitrarily small, but not zero.

Teacher: [Looks on silently, smiling]

Student: So let's define a procedure. Pick some $ε > 0$. Then whenever $\color{red}{0 < |x - a| < δ}$ (in other words, for every $x$ in a $\delta$-neighborhood of $a$), then $\color{lightseagreen}{|f(x) - L| < ε}$.

But I feel like something is missing, because there might not be such a $δ$. For example, if $$f(x) = \begin{cases}-1, & x < 0 \\ 1, & x > 0 \end{cases}$$, then if I pick $ε = 1/2$, the "jump" in $f$ at $x = 0$ has size $2$. So no matter how small I make the $δ$-neighborhood around $a = 0$, this neighborhood will always contain $x$-values that are negative, as well as $x$-values that are positive, which means any such $\delta$-neighborhood will have points where the function has values $1$ and $-1$. It would be impossible to pick a limit $L$ that is simultaneously within $1/2$ unit of $1$ and $-1$, let alone simultaneously arbitrarily close to $1$ and $-1$.

Teacher: Correct. Good job on finding a function that lacks such a $δ$. But why does this function lack such a $δ$?

Student: I don't get what you mean.

Teacher: Remember how we were talking about guaranteeing the (absolute) difference between $f(x)$ and $L$ to be shrunk as small as you please? What consequence does this guarantee have on the $δ$-neighborhood?

Student: Well, there has to be some relationship. As our error tolerance decreases, fewer $x$-values around $a$ will satisfy that tolerance, right? So $δ$ must depend in some way on our choice of $ε$. Well, except in trivial cases like if $f(x)$ is a constant, then any $δ$ works. But the point is the EXISTENCE of a $δ$. It doesn't have to be the largest, or even unique. We merely have to be able to find a sufficiently "small" neighborhood, for which all $x$-values in that neighborhood around $a$ will have function values $f(x)$, within the error tolerance we specified to $L$.

Teacher: Right. So if you were to put all of this together, how would you define the concept of a limit?

Student: I'd say that $$\lim_{x \to a} f(x) = L$$ if, for any $\epsilon > 0$, there exists some $δ > 0$ such that for every $x$ satisfying $\color{red}{0 < |x - a| < δ}$, one also has $\color{lightseagreen}{|f(x) - L| < ε}$.

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  • $\begingroup$ Who is this student who came up with the word "neighbourhood" by themselves?! $\endgroup$
    – Stef
    Commented Feb 13 at 15:17
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An example: For an experiment you need $N$ cells of a certain bacterial species. You cultivate them under ideal conditions in a growth medium so that their number increases exponentially. Let $P(t) = c \cdot a^t$ be the number of bacteria at time $t$. First you answer the question how long you have to wait until you have $N$ bacteria cells. The $\epsilon$-$\delta$-definition can be motivated by asking the question: How accurate do you have to be so that the error in the number of cells is less than $\epsilon$? What happens when due to increased requirements to the experiment you need do decrease the maximal error $\epsilon$?

In the course I would use concrete values for $N$, $\epsilon$ etc.

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