My Favorite One-Liners: Part 75

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

The \delta-\epsilon definition of a limit is often really hard for students to swallow:

\forall \epsilon > 0 \exists \delta > 0 \forall x (0 < |x - c| < \delta \Rightarrow |f(x) - L| < \epsilon)

To make this a little more palatable, I’ll choose a simple specific example, like \lim_{x \to 2} x^2 = 4, or

\forall \epsilon > 0 \exists \delta > 0 \forall x (0 < |x - 2| < \delta \Rightarrow |x^2 - 4| < \epsilon)

I’ll use one of the famous lines from “Annie Get Your Gun”:

Anything you can do, I can do better.

In other words, no matter how small a \delta they give me, I can find an \epsilon that meets the requirements of this limit.

 

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