Limites en Calcul
Comprendre les limites et leur rôle dans le calcul
Interactive Limits Visualization
Welcome to the Limits Explorer
This interactive visualization helps you understand limits and how functions behave as x approaches a specific value. Explore six different function types to see various limit scenarios, including cases where limits exist, don't exist, or are infinite.
What you can explore:
- Continuous functions - See how limits work for smooth functions (f(x) = x²)
- Jump discontinuities - Left and right limits exist but are different
- Removable discontinuities - Limit exists but function is undefined at that point
- Infinite discontinuities - Function approaches infinity as x approaches the point
- Oscillating functions - No limit exists due to infinite oscillation
- Rational functions - Explore limits with removable discontinuities
How to Use This Visualization
Interactive Features:
- • Select Function Type - Choose from 6 different functions to explore various limit scenarios
- • Adjust Approaching x - Use the slider or click on the graph to set the approaching value
- • Toggle Visualizations - Show/hide approaching point, left/right limits, and ε-δ visualization
- • Animate Approach - Watch x approach the limit point a to see the limit concept in action
- • Epsilon-Delta - Explore the formal definition of limits with ε-δ bands
What You'll See:
- • Function Curve (colored) - The original function with proper discontinuity handling
- • Point a (red) - The limit point marked with a vertical line
- • Approaching Point (green) - The point x approaching a, with connection line
- • Left/Right Limits (blue/amber) - Points showing approach from left and right
- • Holes - Empty circles for removable discontinuities
- • Limit Information Panel - Shows left limit, right limit, overall limit, and f(a)
- • ε-δ Visualization - Horizontal (ε) and vertical (δ) bands when enabled
Continuous function - limit exists and equals function value
Click on the graph to set x
Limit Information
Point a: 1.00
Left Limit: limx→a⁻ f(x) = 1.00
Right Limit: limx→a⁺ f(x) = 1.00
Limit: limx→a f(x) = 1.00
f(a): 1.00
Key Concepts
Limit: The value that f(x) approaches as x approaches a, regardless of the actual value at a.
Left Limit: The limit as x approaches a from the left (x → a⁻).
Right Limit: The limit as x approaches a from the right (x → a⁺).
ε-δ Definition: For every ε > 0, there exists δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.
Limit Definition
limx→a f(x) = L
The limit exists if and only if the left-hand limit equals the right-hand limit. If they differ, the limit does not exist (DNE).
Leçons
Unités d'apprentissage individuelles
Leçons à venir