Epsilon and Delta

Making “close” precise

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $\lim_{x \to a} f(x) = L$: for every $\varepsilon > 0$ there's a $\delta > 0$ with $|f(x) - L| < \varepsilon$ whenever $0 < |x - a| < \delta$.
• For a line of slope $m$, $\delta = \varepsilon / |m|$ works.
• The ε–δ definition turned intuition about “closeness” into rigorous mathematics.

Steps

1. A challenge game
2. Win the game
3. A curved graph
4. An unwinnable game
5. A steeper line
6. A lopsided curve