Overview
This lecture introduces the precise (epsilon-delta) definition of a limit in calculus, explains its meaning, and works through formal proofs for linear and quadratic functions using this definition.
Intuitive Concept of Limits
- The limit of f(x) as x approaches a is L if f(x) can get arbitrarily close to L for x values sufficiently close to, but not equal to, a.
- Notation: (\lim_{x \to a} f(x) = L).
Formal Epsilon-Delta Definition
- Let f be defined on an open interval around a (except possibly at a).
- The limit of f(x) as x approaches a is L if, for every epsilon (ε) > 0, there exists delta (δ) > 0 such that whenever (0 < |x - a| < \delta), then (|f(x) - L| < \epsilon).
- The ε–δ definition proves that f(x) can be made as close to L as desired by choosing x close enough to a.
Understanding the Epsilon-Delta Definition
- (|f(x) - L| < \epsilon) means f(x) is within ε units of L (inside the interval (L - \epsilon < f(x) < L + \epsilon)).
- (0 < |x - a| < \delta) means x is within δ units of a but not equal to a.
- For every ε, you must find a δ such that this implication holds.
Example: Linear Function ((\lim_{x \to 2} (3x+1) = 7))
- Given ε > 0, choose δ = ε / 3.
- Prove that if (|x-2| < \delta), then (|3x + 1 - 7| = 3|x-2| < \epsilon).
- Use algebraic manipulation backward from the desired inequality to find δ.
Example: Linear Function ((\lim_{x \to -1} (\frac{1}{3}x + 4) = \frac{11}{3}))
- Given ε > 0, choose δ = 3ε.
- If (|x+1| < δ), then (|\frac{1}{3}x + 4 - \frac{11}{3}| = \frac{1}{3}|x+1| < \epsilon).
Example: Quadratic Function ((\lim_{x \to 2} (2x^2 - 3x - 5) = -3))
- Start with (|2x^2 - 3x - 5 + 3| = |2x^2 - 3x - 2|).
- Factor to get (|2x + 1||x - 2|).
- Choose an interval near 2 (e.g., (1 < x < 3)) to bound (|2x + 1|), then δ = min{1, ε/7}.
- Use a case split based on ε size to formally show the limit holds.
Key Terms & Definitions
- Limit — The value that f(x) approaches as x approaches a.
- Epsilon (ε) — Any positive distance specifying closeness to the limit L.
- Delta (δ) — The corresponding distance from a that ensures f(x) is within ε of L.
- Epsilon-Delta Definition — The formal, precise definition of the limit using ε and δ.
Action Items / Next Steps
- Practice stating the formal definition of a limit.
- Review and attempt similar epsilon-delta proofs for other functions.
- Prepare for test questions requiring the precise definition and basic examples.