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미적분학의 극한

극한 및 미적분학에서의 역할 이해

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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미적분학의 극한 | Maths Learning