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.

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.”

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.

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 $\epsilon$:
Day 1: $\epsilon$ = 1″ Week 2: $\epsilon$ = 0.5″ Month 2: $\epsilon$ = 0.2″

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.

The Upshot

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

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