How Precise Must a Part Be?

A customer wants a piston face with area within ε of the target. How precisely must the radius be cut? That question, with an answer δ for every ε you could demand, is exactly the epsilon-delta definition of a limit.

How it works

The precise definition of a limit

In one sentence

$\lim_{x\to a}f(x) = L$ means: for every $\varepsilon > 0$ there is a $\delta > 0$ such that $0 < |x - a| < \delta$ implies $|f(x) - L| < \varepsilon$ .

General formula

\lim_{{\htmlData{sym=3}{r}}\to {\htmlData{sym=2}{r_0}}}\pi {\htmlData{sym=3}{r}}^2 = \pi {\htmlData{sym=2}{r_0}}^2: \quad \text{for every } {\htmlData{sym=0}{\varepsilon}} > 0 \text{ there is a } {\htmlData{sym=1}{\delta}} > 0 \text{ with } |{\htmlData{sym=3}{r}} - {\htmlData{sym=2}{r_0}}| < {\htmlData{sym=1}{\delta}} \Rightarrow |\pi {\htmlData{sym=3}{r}}^2 - \pi {\htmlData{sym=2}{r_0}}^2| < {\htmlData{sym=0}{\varepsilon}}

$\varepsilon$ is the challenge (how close the output must be), $\delta$ is the response (how close the input must be). A limit exists when every challenge can be met.

What the symbols mean

$\varepsilon$
Area tolerance
How far from the target area is acceptable.
$\delta$
Radius tolerance
How far from the target radius the cut may be.
$r_0$
Target radius
The radius on the drawing.
$r$
Actual radius
What the machine actually cuts.

The tolerance band $|A - A_0| < \varepsilon$ is a horizontal strip around the target area.
We need a vertical strip $|r - r_0| < \delta$ of radii that all land inside that horizontal strip.
Near $r_0$ , $|\pi r^2 - \pi r_0^2| = \pi|r + r_0||r - r_0| \approx 2\pi r_0|r - r_0|$ , so $\delta \approx \varepsilon/(2\pi r_0)$ works.
Demand a tighter $\varepsilon$ and a smaller $\delta$ still works: that's what it means for the limit to exist.

Learn the math

The Precise Definition of a LimitCalc I · Lesson 13, Limits DifferentialsCalc I · Lesson 37, Applications of Derivatives Measurement Error: Accuracy and PrecisionEng Stats · Lesson 1, Engineering Data and Variation Sums of Random Variables and Tolerance Stack-UpEng Stats · Lesson 7, Combining Variation Propagation of ErrorEng Stats · Lesson 8, Combining Variation Tolerance IntervalsEng Stats · Lesson 15, Intervals for Engineering Decisions

Learn the math

The Precise Definition of a LimitCalc I · Lesson 13, Limits

DifferentialsCalc I · Lesson 37, Applications of Derivatives

Measurement Error: Accuracy and PrecisionEng Stats · Lesson 1, Engineering Data and Variation

Sums of Random Variables and Tolerance Stack-UpEng Stats · Lesson 7, Combining Variation

Propagation of ErrorEng Stats · Lesson 8, Combining Variation

Tolerance IntervalsEng Stats · Lesson 15, Intervals for Engineering Decisions