Límites en Cálculo
Comprender los límites y su papel en el cálculo
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).
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