Select a function, set a (the point to approach), then click Capture. Try different values of ε to see how δ changes.
f(x) = sin(x)/x
Recorded Data
a
L⁻
L⁺
Limit
ε
δ
Continuity
Set a and ε then click Capture
Live Graph (δ vs ε)
Record 2+ points to see graph
What To Explore
For sin(x)/x as x→0: does the limit exist even though f(0) is undefined?
For |x|/x: left and right limits — are they equal?
Decrease ε: what happens to δ? This is the formal ε-δ definition of a limit.
EQUATION LAB — Limits & Continuity
Symbols
LThe limit value (if it exists)
aThe point being approached
εEpsilon — how close we need f(x) to be to L
δDelta — how close x must be to a
L⁻Left-hand limit (x → a⁻)
L⁺Right-hand limit (x → a⁺)
Formal Definitions
Epsilon-Delta Definition
The limit L exists if for every ε > 0 there is a δ > 0 such that whenever 0 < |x − a| < δ, we have |f(x) − L| < ε. The yellow band shows ε; the blue band shows δ.
Limit Existence
The two-sided limit exists only when both one-sided limits exist and are equal.
Continuity at a Point
Three conditions must hold: f(a) defined, limit exists, and limit equals f(a).