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A.2 Tolerance & Pancake Story

To begin making our definition of “limit” from the preceding Topic more precise, on this screen we will introduce the notion of “tolerance,” or “error,” in the output of a process, and introduce a way to visualize the connection between an allowable tolerance and the associated difference in inputs.

Recall that working definition of “limit”:

The limit of the function $f(x)$ as x approaches a, written

\[\lim_{x \to a} f(x) = L\]

is a number L (if one exists) such that $f(x)$ is as close to L as we want
whenever x is sufficiently close to a.

 
As we noted, the words “as close to” and “sufficiently close” are not precise. Let’s dive in a little deeper and think first about what it means for $f(x)$ to be “as close to L as we want.”

Manufacturing specs always specify an allowable ‘tolerance’
While you are probably not used to thinking about a function’s input or output being “close to” one value or another in a math class, you are used to such thinking in some everyday contexts. For instance, any item that is manufactured inevitably has a bit of “error” in its production. For this reason, if the part will become part of a larger product, typically the design team specifies both the desired “correct size” of the part, along with the allowable tolerance that specifies the allowable error. The size of that tolerance depends on how the part will be used: for example, while a bolt used as part of a newly-built house may need to be within only ±0.1 cm of a stated 1.0-cm length, a bolt used as part of a SpaceX Falcon Rocket may need to be manufactured to within the narrow range of $1.0000 \pm 0.0001$ cm.

Pancakes and tolerance
To explore these ideas more deeply, we’re going to use what we hope is a more familiar everyday context: making pancakes. We know that this scenario is a little goofy, and the data we’re going to present in the scenario are completely idealized and ignore many real-world factors. But previous students who have worked through similar materials developed a solid and necessary understanding of super-critical ideas [Ref], and so we’re using this simple scenario to lay a good foundation for our crucial work ahead.


 

Before we leave the pancake scenario, let’s introduce a little notation that will make it easier for us to link the groundwork we laid here with the work that is to come.

Specifically, imagine that your boss tells you that it’s too many words to say, “Today the tolerance on your pancake size is ± 1/2 inch.” Instead he’s going to give you a single number called epsilon (written $\epsilon$, the lower-case Greek letter “e”), and say “Today your $\epsilon$ = 1/2 inch,” and you have to take it from there. He’s chosen epsilon, $\epsilon,$ because (1) he’s Greek and (2) this value represents the allowable error in your pancake size.

$\epsilon$: allowable output error.
$\delta$: difference in input value.
Furthermore, he expects you to then report back how precisely you must pour your batter to be within his tolerance $\epsilon.$ And he wants to hear back just a single number of tablespoons you can be “off by” in your pouring, signified by lower-case delta ($\delta$, the lower-case Greek letter “d”). For example, when he set $\epsilon$ = 1/2 inch, you determined that you must pour within $\delta$ = 1 tablespoon of 14 tablespoons. He’s chosen $\delta$ because (1) he’s still Greek and (2) this value is the maximum difference you can have between the the amount of batter you pour and the ideal 14 tablespoons.

Activity 2: Pancake scenario with $\epsilon$ and $\delta$

Below is another set of linked graphs, with the upper graph focused on the output pancake-size, and the lower graph focused on the input batter-amount. Initially the value of $\epsilon$ is set to what it was on Day 1 of your time at Perfect Pancakes, $\epsilon$ = 1″. But you can “be the boss” and change the value of $\epsilon$ by using the buttons beneath the lower calculator.

Please verify, for each value of $\epsilon$ available, that you can find at least one value of $\delta$ such that the function’s output falls within the required range.

Desmos Tip: To zoom only vertically on a laptop: place your cursor near the the upper graph’s vertical axis, and then hold down the shift-key while scrolling vertically.

Focus on Pancake Size:
The size-range must be within the Boss’s specified $\pm \epsilon$
Day 1 the Boss says $\epsilon = 1$”
Focus on Batter Used:
Within what $\delta$ must you scoop so the pancakes are in the desired range?

Use the slider to set how precisely you pour the batter:

Currently you are pouring within $\delta = \pm$ 2.4 T of the ideal 14 T.

Be the Boss and select the value of $\epsilon$:
    

 
Please make sure you’ve verified that you can find at least one value of $\delta$ such that the resulting pancakes all fall within the required $\pm \epsilon$ output range. More importantly, please make sure you understand how setting $\epsilon$ (focusing on the output’s variation) determines how close ($\pm \delta$) you must be to the “ideal” input value. We’ll tie these ideas directly into developing the formal definition of “limit” below.

We’ll say it again: we know the scenario of the Pancake Story is a little goofy. But we also imagine it’s done the necessary job to link epsilon (tolerance, or output error) and delta (difference in input from the ideal 14 T) to our working definition of limit, which we do now with some simple additions to the last line:

\[\text{The limit of the function $f(x)$ as $x$ approaches $a$, written} \] \[\lim_{x \to a} f(x) = L \] \[\text{is a number $L$ (if one exists)}\] \[\text{such that $f(x)$ is $\underbrace{\text{as close to}}_{\Large{\text{within } \pm
\epsilon} \text{ of}}$ $L$ as we want whenever $x$ is $\underbrace{\text{sufficiently close to}}_{\Large{\text{within } \pm \delta \text{ of}}}$ $a.$ }\]

On the next screen, we’ll return to the world of mathematical functions and use our understanding of $\delta$ and $\epsilon$ to re-examine $\displaystyle{\lim_{x \to 2}\dfrac{x^2\, -\, 4}{x\,-\,2}}$ from the preceding screen.


We hope this screen has done its job of introducing the ideas of epsilon $(\epsilon)$ and delta $(\delta)$ in an understandable way. What’s your experience? Please let us know over on the Forum!


The Upshot

  1. $\epsilon$: epsilon, the allowable output error, or tolerance.
    $\delta$: delta, the difference in input value.
  2. Epsilon and delta tie directly into our working definition of “limit”:
    \[\text{The limit of the function $f(x)$ as $x$ approaches $a$, written} \] \[\lim_{x \to a} f(x) = L \] \[\text{is a number $L$ (if one exists)}\] \[\text{such that $f(x)$ is $\underbrace{\text{as close to}}_{\Large{\text{within } \pm
    \epsilon} \text{ of}}$ $L$ as we want whenever $x$ is $\underbrace{\text{sufficiently close to}}_{\Large{\text{within } \pm \delta \text{ of}}}$ $a.$ }\]


Reference:
Adiredja, A. P. (2019). “The pancake story and the epsilon-delta definition.” PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. https://doi.org/10.1080/10511970.2019.1669231