Finding Deltas Algebraically


Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.


f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03

Verified step by step guidance

1. Identify the function and the given values: f(x) = mx, L = 2m, c = 2, and ε = 0.03.
2. Set up the inequality |f(x) - L| < ε, which becomes |mx - 2m| < 0.03.
3. Simplify the inequality: |m(x - 2)| < 0.03. This implies -0.03 < m(x - 2) < 0.03.
4. Divide the entire inequality by m (since m > 0, the direction of the inequality remains unchanged): -0.03/m < x - 2 < 0.03/m.
5. Determine the largest open interval about c = 2: The interval is (2 - 0.03/m, 2 + 0.03/m). Choose δ = 0.03/m to ensure that 0 < |x - 2| < δ implies |f(x) - L| < ε.

Key Concepts

Limit Definition

In calculus, the limit of a function describes the behavior of that function as its input approaches a certain value. Specifically, for a function f(x) to approach a limit L as x approaches c, the values of f(x) must get arbitrarily close to L when x is sufficiently close to c. This concept is foundational for understanding continuity and the formal definition of limits.

Epsilon-Delta Definition of a Limit

The epsilon-delta definition formalizes the concept of limits in calculus. It states that for every ε (epsilon) greater than 0, there exists a δ (delta) such that if the distance between x and c is less than δ (but not zero), then the distance between f(x) and L is less than ε. This precise definition is crucial for proving the existence of limits and understanding their properties.

Open Intervals

An open interval around a point c, denoted as (c - δ, c + δ), includes all numbers between c - δ and c + δ but does not include the endpoints. In the context of limits, identifying the largest open interval where the limit condition holds is essential for determining the range of x values that satisfy the inequality |f(x) - L| < ε. This concept helps in visualizing the behavior of functions near specific points.