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Continuity and Limits: Epsilon-Delta Visualization for Calculus

Continuity and Limits: Epsilon-Delta Visualization for Calculus
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2012-06-07 10:46
FunctionCalculusLimitGraphing
This GeoGebra applet illustrates foundational ideas for a beginning calculus course, focusing on points of discontinuity and removable discontinuities. It provides a dynamic visualization of limits and continuity using the epsilon-delta (ε-δ) framework. The construction relies on GeoGebra commands such as Function to plot the curve, Point to mark significant locations, OrthogonalLine and Line to construct local neighborhoods, Intersect to determine boundary points, Polygon to highlight relevant regions, and Text for annotations. Key objects include the function f, labeled points B through V, and the parameters ε and δ. The example is well suited for introductory calculus students and instructors who need an interactive tool to demonstrate how limits define continuity and how removable discontinuities appear graphically.

GeoGebra Commands & Parameters

  1. Function(2 * x + 3, -3, 0) → g
  2. Function((x + 9) * (x + 7) / ((x + 9) * (x + 7)), -10000, -7) → i
  3. Point(g) → B
  4. Function(1 / 4 * (x^(2) + 1) * x / (x * (x - 3)), 0, 10000) → f
  5. Function((-x - 6) * (x + 7) / (x + 7), -7, -3) → h
  6. Point(h) → E
  7. OrthogonalLine(K, xAxis) → a
  8. OrthogonalLine(L, xAxis) → c
  9. Polygon(M, N, O, P) → poly1, m, n, o, p
  10. OrthogonalLine(Q, yAxis) → d
  11. OrthogonalLine(R, yAxis) → e
  12. Polygon(S, T, U, V) → poly2, s, t, u, v
  13. Intersect(c, m) → A
  14. Intersect(c, m) → W
  15. Intersect(c, m) → Z
  16. Line(E, B) → k
  17. Intersect(k, c) → A_1
  18. OrthogonalLine(B, yAxis) → l
  19. OrthogonalLine(A_1, yAxis) → q
  20. Intersect(yAxis, q) → B_1
  21. Intersect(l, c) → C_1
  22. Polygon(A_1, B_1, B, C_1) → poly3, a_1, b_1, b_2, c_1
  23. Intersect(a, o) → D_1
  24. Intersect(a, o) → E_1
  25. Intersect(a, o) → F_1
  26. Intersect(a, o) → G_1
  27. OrthogonalLine(K, o) → r
  28. Intersect(a, o) → H_1
  29. Intersect(a, o) → I_1
  30. Intersect(a, o) → J_1
  31. Intersect(f, a, (0.15, -0.08969298245614034)) → K_1
  32. OrthogonalLine(K_1, yAxis) → d_1
  33. Intersect(yAxis, d_1) → L_1
  34. Polygon((0, 0), L_1, K_1, K) → poly5, e_1, f_1, g_1, h_1
  35. Text("Discontinuity at x = -9.") → text3
  36. Text("Discontinuity at x = -7.") → text4
  37. Text("Discontinuity at x = 0.") → text5
  38. Text("Removable") → text6
  39. Text("Removable") → text7
  40. Text("Nonremovable") → text8
  41. Text("Drag b on the blue slider bar to change, x, the input value.") → text10
  42. Text("What this Applet Does") → text12
  43. Text("Nonremovable") → text13
  44. Text("Discontinuity at x = 0.") → text14

AI Drawing Prompt Reference

Create a calculus teaching graph illustrating limits and continuity with an epsilon-delta visualization. The graph features a piecewise-defined curve with discontinuities at x = -9, x = -7, and x = 0, where the first two are removable and the last is nonremovable. Use perpendicular lines to construct local neighborhoods, determine boundary points via intersections, and highlight relevant regions with polygons. Label key points on the axes, add text annotations explaining each discontinuity type, and include a note prompting users to drag a slider to change the input value and observe the effect.
The prompt above was compiled from the original drawing commands and is for reference only.