What is the epsilon-delta definition of a limit?

The epsilon-delta definition states that lim(x→c) f(x) = L if for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - c| < δ, we have |f(x) - L| < ε. In other words, we can make f(x) as close to L as we want (ε-close) by making x sufficiently close to c (δ-close). The visualization shows these bands graphically.

What types of discontinuities exist?

There are four main types: (1) Removable — the limit exists but f(c) is either undefined or differs from the limit (a ‘hole’). (2) Jump — the left-hand and right-hand limits both exist but are not equal. (3) Infinite — the function approaches positive or negative infinity (a vertical asymptote). (4) Oscillating — the function oscillates too rapidly near c for any limit to exist, such as sin(1/x) near x = 0.

When does a limit not exist?

A limit does not exist when: (1) the left-hand and right-hand limits are different (jump discontinuity), (2) the function approaches infinity (infinite discontinuity), (3) the function oscillates without settling on a value (oscillating discontinuity), or (4) the function behaves erratically near the point. For a limit to exist, both one-sided limits must exist and be equal.

Limits & Continuity Visualizer

Q: What types of discontinuities exist?

There are four main types: (1) Removable — the limit exists but f(c) is either undefined or differs from the limit (a ‘hole’). (2) Jump — the left-hand and right-hand limits both exist but are not equal. (3) Infinite — the function approaches positive or negative infinity (a vertical asymptote). (4) Oscillating — the function oscillates too rapidly near c for any limit to exist, such as sin(1/x) near x = 0.

Q: When does a limit not exist?

A limit does not exist when: (1) the left-hand and right-hand limits are different (jump discontinuity), (2) the function approaches infinity (infinite discontinuity), (3) the function oscillates without settling on a value (oscillating discontinuity), or (4) the function behaves erratically near the point. For a limit to exist, both one-sided limits must exist and be equal.

Explore removable, jump, infinite, and oscillating discontinuities. Watch two dots approach the limit point and see epsilon-delta bands in action.

Limit Explorer

Presets

Approach x = c = 1.0

1.0

Epsilon ε = 0.5

0.5

Show ε-δ bands

Properties

Type	--
Function	--
Left limit	--
Right limit	--
Limit	--
Continuous?	--

The Math Behind It

Limits

Definition: lim_x→c f(x) = L means f(x) gets arbitrarily close to L as x approaches c
Left-hand limit: lim_x→c⁻ f(x) — approaching from the left
Right-hand limit: lim_x→c⁺ f(x) — approaching from the right
The limit exists if and only if left = right
ε-δ definition: for every ε > 0, there exists δ > 0 such that |x - c| < δ implies |f(x) - L| < ε

Types of Discontinuity

Removable: a “hole” in the graph — limit exists but f(c) is undefined or ≠ L
Jump: left-hand limit ≠ right-hand limit — the function “jumps”
Infinite: f(x) → ±∞ as x → c — vertical asymptote
Oscillating: f(x) = sin(1/x) near 0 — oscillates too wildly for a limit to exist
A function is continuous at c if f(c) = lim_x→c f(x)

Limits & Continuity Visualizer

Limit Explorer

Properties

The Math Behind It

Limits

Types of Discontinuity

Related Visualizations

Derivative Visualizer

Taylor Series

All Visualizations

You're crushing it!