Good ways of explaining the idea of epsilon-delta limits to bio & chem majors?

Question

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.

Answer 1

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!

Answer 2

(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.

Answer 3

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.