Epsilon — Definition, Formula & Examples

Epsilon (ε) is a Greek letter that represents an arbitrarily small positive number, most commonly used in the formal definition of limits. It quantifies how close a function's output must be to the limit value.

In the epsilon-delta definition of a limit, $\varepsilon$ denotes a positive real number such that for every $\varepsilon > 0$ , there exists a corresponding $\delta > 0$ satisfying the condition $0 < |x - c| < \delta \implies |f(x) - L| < \varepsilon$ , where $L$ is the limit of $f(x)$ as $x \to c$ .

Key Formula

\forall\, \varepsilon > 0,\; \exists\, \delta > 0 \text{ such that } 0 < |x - c| < \delta \implies |f(x) - L| < \varepsilon

Where:

$\varepsilon$ = An arbitrarily small positive number bounding the distance from f(x) to L
$\delta$ = A positive number bounding the distance from x to c
$L$ = The limit value of f(x) as x approaches c
$c$ = The point that x approaches

How It Works

In an epsilon-delta proof, your goal is to show that no matter how small someone chooses

\varepsilon

, you can always find a

\delta

that keeps

f(x)

within

\varepsilon

of the limit

L

. Think of

\varepsilon

as a tolerance on the output: it defines a band

(L - \varepsilon,\; L + \varepsilon)

around the limit. Your job is to find a corresponding input window

(c - \delta,\; c + \delta)

that guarantees

f(x)

stays inside that band. Because

\varepsilon

is arbitrary, the proof must work for every positive value, no matter how tiny.

Worked Example

Problem: Use the epsilon-delta definition to prove that the limit of f(x) = 3x + 1 as x → 2 is 7.

Set up the epsilon condition: We need |f(x) - L| < ε, which becomes |3x + 1 - 7| < ε.

|3x - 6| < \varepsilon

Factor and relate to |x - c|: Factor out 3 to connect the expression to |x - 2|.

3|x - 2| < \varepsilon \implies |x - 2| < \frac{\varepsilon}{3}

Choose delta: Set δ = ε/3. Then whenever 0 < |x - 2| < δ, we get |3x + 1 - 7| = 3|x - 2| < 3·(ε/3) = ε.

\delta = \frac{\varepsilon}{3}

Answer: For every ε > 0, choosing δ = ε/3 guarantees |f(x) - 7| < ε, proving the limit is 7.

Why It Matters

Epsilon-delta proofs are the rigorous foundation of calculus, required in college-level real analysis and advanced calculus courses. Mastering this notation is essential for proving continuity, differentiability, and convergence — concepts that underpin engineering, physics, and computer science applications.

Common Mistakes

Mistake: Choosing a specific value for ε (like ε = 0.01) instead of keeping it arbitrary.

Correction: The proof must work for every ε > 0. Your δ should be expressed as a function of ε (e.g., δ = ε/3), not tied to a single number.