Overview
This lecture introduces and explains the precise (epsilon-delta) definition of a limit in calculus, focusing on the formal notion of "closeness" as it pertains to functions.
The Precise Definition of a Limit
- The limit of f(x) as x approaches a is L if, for every ε > 0, there exists δ > 0, such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
- f(x) does not need to be defined at x = a for the limit to exist.
- The definition formalizes the intuitive idea that "values of f(x) get close to L when x is close to a."
Understanding Absolute Value Inequalities
- |x - a| < δ describes all x within δ units of a, but not equal to a.
- Making δ smaller means x is even closer to a.
- Similarly, |f(x) - L| < ε means f(x) is within ε units of L.
- Smaller ε requires the function values to be closer to the limit value.
Interpreting Epsilon and Delta
- Given any ε (closeness required on the y-axis), you can find a δ (closeness on the x-axis) so that the definition holds.
- Smaller ε values mean we must select smaller δ intervals to ensure the function values stay sufficiently close to L.
- The process captures and quantifies the idea of "closeness" for limits.
Example Explanation (without proof)
- For the statement lim_{x→7} f(x) = 4, as x gets close to 7 (from either side), f(x) gets close to 4.
- The function need not be defined at x = 7 for the limit to exist.
Key Terms & Definitions
- Limit — The value that f(x) approaches as x approaches a specific point a.
- Epsilon (ε) — Any positive number representing desired closeness to the limit value on the y-axis.
- Delta (δ) — Any positive number representing required closeness to the point a on the x-axis.
- Absolute Value Inequality — An expression like |x - a| < δ describing how close x is to a.
Action Items / Next Steps
- Review absolute value inequalities and their solutions.
- Watch the upcoming video where specific limit examples are proven using the precise definition.