In the epsilon-delta definition game, which quantity is chosen first, epsilon or delta?

Conditions

• The definition is viewed as a challenge or game.
• $\epsilon$ is the independent variable in the logical quantification.
• $\delta$ depends on the chosen $\epsilon$.

Reasoning, step by step

1. A challenger selects an arbitrary $\epsilon > 0$.
2. The responder must find a $\delta > 0$ that works for that specific $\epsilon$.
3. The validity of the limit depends on the responder's ability to find such a $\delta$ for *every* possible $\epsilon$.
4. Conclude that $\epsilon$ precedes $\delta$ in the logical order.

Example

The board explicitly writes 'give $\epsilon > 0$' followed by 'They'll give a $\delta$'. The speaker describes this as: 'you define what you want by giving me an epsilon... I can get you as close as you want... by giving you a range around the point that x is approaching'.

Common misconceptions

• Thinking that $\delta$ is chosen first and $\epsilon$ follows.
• Believing that $\delta$ is independent of $\epsilon$; in fact, $\delta$ is a function of $\epsilon$.
• Confusing the temporal order of drawing on the graph with the logical order of the definition.

Watch the explanation

Explore next

• Why must delta depend on epsilon?
• What happens if you try to choose delta first?
• How does this order reflect the universal and existential quantifiers?