The epsilon-delta definition gives a precise meaning to the statement that f(x) approaches L as x approaches a. It replaces the informal idea of getting close with a measurable challenge and response. The epsilon value measures how close the output f(x) must be to L, while delta measures how close the input x must be to a.
This definition matters because it is the foundation for continuity, derivatives, and rigorous calculus proofs.
The definition says that for every epsilon greater than 0, there must be a delta greater than 0 such that if 0 < |x - a| < delta, then |f(x) - L| < epsilon. The condition 0 < |x - a| means x can be close to a without needing to equal a. In a graph, epsilon forms a horizontal band around y = L, and delta forms a vertical band around x = a.
A proof shows how to choose delta based on a given epsilon so the curve stays inside the epsilon band whenever x is inside the delta band.
Key Facts
- lim x -> a f(x) = L means for every epsilon > 0, there exists delta > 0 such that 0 < |x - a| < delta implies |f(x) - L| < epsilon.
- Epsilon controls vertical closeness: |f(x) - L| < epsilon.
- Delta controls horizontal closeness: |x - a| < delta.
- The condition 0 < |x - a| excludes x = a because the limit depends on nearby values, not necessarily the value at a.
- For f(x) = mx + b, lim x -> a f(x) = ma + b can be proved using |f(x) - L| = |m||x - a|.
- A useful choice in linear examples is delta = epsilon / |m| when m is not 0.
Vocabulary
- Limit
- A limit is the value that a function approaches as the input approaches a specified number.
- Epsilon
- Epsilon is a positive number that represents the allowed distance between f(x) and the proposed limit L.
- Delta
- Delta is a positive number that represents how close x must be to a to force f(x) within epsilon of L.
- Punctured neighborhood
- A punctured neighborhood of a is the set of x-values close to a but not equal to a.
- Continuity
- A function is continuous at a when its limit as x approaches a equals its actual value f(a).
Common Mistakes to Avoid
- Choosing epsilon after delta is given: this is wrong because the definition requires a delta response for every epsilon challenge.
- Forgetting 0 < |x - a|: this is wrong because limits describe behavior near a and do not require the function to be defined at a.
- Treating delta as a fixed universal number: this is wrong because delta often depends on the chosen epsilon and on the function.
- Proving only one epsilon value: this is wrong because the definition must work for every positive epsilon, no matter how small.
Practice Questions
- 1 Prove using the epsilon-delta definition that lim x -> 3 (2x + 1) = 7. Find a formula for delta in terms of epsilon.
- 2 For f(x) = 5x - 4 and a = 2, find a delta that guarantees |f(x) - 6| < 0.01 whenever 0 < |x - 2| < delta.
- 3 Explain why a function can have lim x -> a f(x) = L even if f(a) is undefined or f(a) is not equal to L.