Interactive Lab · Limits
Epsilon–Delta Challenge
Turn the formal definition into something you can see: meet an output demand by choosing an input neighborhood that works for every nearby point.
Live experiment
Choose δ as a factor of ε
Set the output tolerance, then choose the factor k in δ = kε. The graph checks every nearby point automatically.
When you are ready, turn Challenge on and generate a new example. The one-sided constraints stay hidden while you determine a working δ factor; Show solution reveals them when you want to check your reasoning.
0 < |x − a| < δimplies|f(x) − L| < ε
Quadratic explorationf(x) = x², a = 2, L = 4
ε-bandδ-neighborhood
Output band3.00 < f(x) < 5.00
Input neighborhood1.85 < x < 2.15, x ≠ 2
One-sided constraintsδL = —δR = —
Which single δ guarantees both sides?
Connect the experiment to the proof
The graph suggests the rule. The definition proves it.
Once you can see why the smaller one-sided constraint controls the symmetric neighborhood, return to the lesson and express the same reasoning with quantified inequalities.
Read The Formal Definition of a Limit →