Epsilon-delta definition
The limit of f(x) at a is L if, for every tolerance epsilon > 0, there is a distance delta > 0 so that x within delta of a gives f(x) within epsilon of L.
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Analogy
A dart game: however small the bullseye (epsilon) is, you can find a steady-enough aim (delta) to hit it.
Example
|f(x) - 7| = |2x - 6| = 2|x - 3|.We want this less than epsilon.Choose delta = epsilon / 2.Then |x - 3| < delta gives |f(x) - 7| < epsilon.
In practice
It is the exact definition behind all limit rules, and the starting point of formal analysis.
Practice
Flashcards
- In epsilon-delta for 2x + 1 near 3, which delta works?delta = epsilon / 2.
Questions people ask
How do I choose delta in an epsilon-delta proof?
Start from |f(x) - L| and rewrite it in terms of |x - a|. Then find a delta that forces it below epsilon. For 2x + 1 near 3 the difference is 2|x - 3|, so delta = epsilon over 2 works. Check by substituting back.
Why do we need such a formal definition of limit?
Words like 'gets close' are vague and can fail on tricky functions. The epsilon-delta definition is exact, so proofs can be checked line by line. It also lets mathematicians prove the limit laws and the squeeze theorem rather than just trusting them.