Lesson 1.2: Formal Definition of a Limit (The ε-δ Defintion)

As I have alluded to in the previous post,lim_x→x0 ƒ(x) = y₀ means that as x approaches x₀, ƒ(x) gets arbitrarily close to y₀. The reason this is not considered a rigorous definition is because the phrases, "as x approaches x₀," and "ƒ(x) gets arbitrarily close to y₀" have not been formally defined. We should therefore, officially present a more rigorous definition for the limit called the ε-δ definition:

Definition 1.1: Limit


Let ƒ(x) be a function defined on an open interval containing x₀, except possibly at x₀ The statement

means that for each ε (Greek letter epsilon) > 0, there exists δ (Greek letter delta) > 0 such that

0 < |x - x₀| < δ implies that |ƒ(x - y₀| < ε. ■

To see what this definition means, we must deconstruct each inequality. The first inequality has two parts. The first part, 0 < |x - x₀| simply means x ≠ x₀.

The second part of the first inequality is |x - x₀| < δ and can be rewritten as -δ < x - x₀ < δ. Adding δ to all three sides, we get

In other words, x lies in the interval (x₀ - δ, x₀ + δ). Similarly, the second inequality can be rearranged and written as

So,the second inequality means that ƒ(x) lies in the interval (y₀ - ε, y₀ + ε). With this, we an get a clearer interpretation of the ε-δ definition:

Interpretation of Definition 1.1


Let ƒ(x) be a function defined on an open interval containing x₀, except possibly at x₀ The statement

means that for each open interval about y_0, I_y = (y₀ - ε, y₀ + ε), there exists an interval about x_0,  I_x = (x₀ - δ, x₀ + δ) such that

x₀ ∈ I_x implies that ƒ(x) ∈ I_y. ■

The symbol ∈ means "is contained in," or "is an element of." The following image is a graphical representation of the ε-δ definition. Notice that y₀ ± ε does not necessarily correspond to ƒ(x₀ ± δ); although if y₀ ± ε ≠ ƒ(x₀ ± δ), then it is necessary that the interval (y₀ - ε, y₀ + ε) must contain the interval (ƒ(x₀ - δ), ƒ(x₀ + δ)). In other words, (ƒ(x₀ - δ), ƒ(x₀ + δ)) ⊆ (y₀ - ε, y₀ + ε).