Limits & Continuity Lab
Investigate the formal epsilon-delta definition of limits, visualize the squeeze theorem in action, and classify discontinuities by testing the three conditions for continuity at a point.
Guided Experiment: Epsilon-Delta Exploration
For the function sin(x)/x near x=0, what is the relationship between ε and δ? Can you always find a δ that works for any given ε?
Write your hypothesis in the Lab Report panel, then click Next.
Controls
Results
ε = 0.500, δ = 0.500
Worst error: 0.041149
δ_max ≈ 1.8955
Data Table
(0 rows)| # | Trial | Mode | Function | Point | Limit | ε | δ | Satisfied |
|---|
Reference Guide
Epsilon-Delta Definition
The limit of f(x) as x approaches a equals L if for every ε greater than 0 there exists a δ greater than 0 such that
Geometrically, for any horizontal band of width 2ε around L, you can find a vertical band of width 2δ around a that keeps the graph inside the box.
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near a, and both g and h approach the same limit L, then f must also approach L.
This is useful for oscillating functions like x²sin(1/x) that are hard to evaluate directly.
Continuity at a Point
A function is continuous at x = a if all three conditions hold
- f(a) is defined
- limx→a f(x) exists
- limx→a f(x) = f(a)
A failure of condition 2 (one-sided limits differ) gives a jump discontinuity. A failure of condition 3 gives a removable discontinuity.
Intermediate Value Theorem
If f is continuous on [a, b] and c is any value between f(a) and f(b), then there exists at least one point x in (a, b) where f(x) = c.
This guarantees that continuous functions cannot "skip" values. It is the theoretical basis for the bisection method of root finding.