The epsilon–delta definition of a limit

The epsilon–delta definition says that lim_x→a f(x) = L when, for every tolerance ε > 0, there is a distance δ > 0 such that every x within δ of a (other than a itself) gives an f(x) within ε of L.

That one sentence is how every modern textbook defines a limit^[1]. It is short, and it is famously hard to read the first time, because it packs two quantifiers ("for every", "there is") into a single line and asks you to keep them in order. This page takes it apart with a figure you can operate, works two proofs from start to finish, and shows what failure looks like. For the intuitive idea of a limit, read the Limits chapter first; this page assumes you know roughly what "approaches" means and want the precise version.

The definition, symbol by symbol§

Read the two inequalities as distances. |f(x) − L| < ε says that the output is within ε of the claimed limit: on a graph, the point lies inside a horizontal band of half-width ε around the height L. The condition 0 < |x − a| < δ says that the input is within δ of a but not equal to it: a vertical strip of half-width δ around a, with the line x = a itself removed. The value f(a) never enters, which is why a function with a hole can still have a limit there.

The order of the quantifiers is the whole point. ε comes first and is arbitrary; δ comes second and may depend on ε. The easiest way to keep this straight is to treat the definition as a game between two players. A challenger names a tolerance ε, as small as they like. You answer with a δ. You win the round if the graph stays inside the ε-band everywhere in the δ-strip. The limit is L exactly when you can win every round, whatever ε the challenger picks^[2].

A worked example: x² near 2§

Claim: lim_x→2 x² = 4. Start with numbers. If the challenger picks ε = 0.5, the band runs from 3.5 to 4.5. The parabola leaves it at x = √3.5 ≈ 1.8708 on the left and at x = √4.5 ≈ 2.1213 on the right. The right side is closer to 2, because the curve is steeper there, so the widest δ that works is √4.5 − 2 ≈ 0.1213. If the challenger tightens to ε = 0.01, the same reasoning gives δ = √4.01 − 2 ≈ 0.0025. In general the best δ is √(4 + ε) − 2, which shrinks as ε shrinks but stays positive. You can always answer.

A proof does not need the best δ, only one that works, and it is usually written with an easier formula. Here is the standard argument^[1]. Factor the distance you must control:

The first factor is what δ controls. The second factor needs a bound. Agree in advance that δ will be at most 1; then x lies between 1 and 3, so |x + 2| < 5. Now choose δ = min(1, ε/5). If 0 < |x − 2| < δ, then |x² − 4| < δ · 5 ≤ (ε/5) · 5 = ε. That is the entire proof. For ε = 0.5 it hands you δ = 0.1, a little smaller than the best possible 0.1213, which is fine: any δ below the best one also wins.

The figure's readout follows the logic of the definition rather than the picture. When a δ exists for a wrong L, it says so and tells you the ε that defeats the claim: for a claimed L, that is the larger of its distances to the two one-sided limits. When the claimed L is the true limit it says that every ε has an answer, which is a mathematical fact about these four examples, not something a computer can confirm by sampling.

A linear function: δ = ε/3§

Lines make the cleanest examples, because the factor you have to bound is a constant. Claim: lim_x→1 (3x + 1) = 4. The distance to control is |3x + 1 − 4| = 3|x − 1|. That is less than ε exactly when |x − 1| < ε/3, so δ = ε/3 works, and it is the best possible. With ε = 0.3, δ = 0.1. The slope sets the exchange rate: a line three times as steep needs a strip three times as narrow.

How to write an ε–δ proof§

Almost every introductory proof follows the same four steps, and the work is done in the first two, on scrap paper^[2]:

1. Write |f(x) − L| and factor out |x − a|, as in equation (2).
2. Bound whatever multiplies it, usually by first insisting that δ ≤ 1 so x stays in a known interval.
3. Set δ = min(1, ε/M), where M is that bound.
4. Write the proof forwards: "Let ε > 0. Choose δ = …. If 0 < |x − a| < δ, then … < ε."

The finished proof reads backwards from how you found it, which is one reason it looks unmotivated on a blackboard. The scrap-paper search for δ is the real mathematics.

What failure looks like§

Switch the figure to the floor function near 1. Just left of 1 the value is 0; just right it is 1. Whatever L you claim, it is at least 0.5 away from one of those two values, so any ε of 0.5 or less defeats it: no band that thin can hold both sides. That is the definition's way of saying the limit does not exist. A wrong claim about a function that does have a limit fails the same way, only the defeating ε is the distance between the claim and the truth. sin(1/x) near 0 fails for every L once ε is below 1, because it sweeps through the whole interval from −1 to 1 in every strip, however narrow. The last example in the figure, x sin(1/x), wiggles just as often but is squeezed toward 0 by the envelope ±x, so every ε has its δ (δ = ε works).

Where the definition came from§

Newton and Leibniz used limits without defining them, and for a century and a half calculus ran on intuition about quantities that become "infinitely small". Augustin-Louis Cauchy's textbooks of the 1820s recast the subject in terms of limits and inequalities, and Judith Grabiner has argued that his proofs already use the logic of ε and δ, even where his words still speak of variables approaching values^[3]. How much of the modern form is Cauchy's is still debated among historians. The fully static version, with no motion at all, only the two quantifiers, is usually credited to Karl Weierstrass's Berlin lectures from the 1850s onward^[4]. The definition removed motion from the idea of a limit: nothing approaches anything; there are only tolerances and guarantees.

For the intuitive picture, one-sided limits and a calculator that shows its evidence, go to Limits. The same pattern of "for every tolerance, a guarantee" returns in the chapter on the derivative, where the derivative is defined as a limit of slopes.