∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 1 Theory

Limits, Continuity & The Epsilon-Delta Definition

Rigorous ε-δ corridors, one-sided limits, and removable discontinuities

Adjust ε and δ to see how the vertical output corridor traps the function inside the horizontal input window around x₀.

1. The Geometry of Epsilon and Delta

A limit L is the value that f(x) approaches as x gets arbitrarily close to x₀. The ε-δ definition guarantees that no matter how narrow an error tolerance ε > 0 you demand around L, there exists a distance δ > 0 around x₀ such that every point within δ lands inside ε.

0<∣x−x0∣<δ  ⟹  ∣f(x)−L∣<ε0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon

Quick Reference & Cheat Sheet

Removable Discontinuity Rule
lim⁡x→ax2−a2x−a=lim⁡x→a(x+a)=2a\lim_{x \to a} \frac{x^2 - a^2}{x - a} = \lim_{x \to a} (x + a) = 2a
A single missing point (hole) in an otherwise smooth continuous curve.