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Limits and Continuity by Prof. Caine

Limits and Continuity by Prof. Caine
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2016-08-03 00:20
FunctionCalculusLimitGeometric ConstructionGraphing
This example graphically illustrates epsilon-delta definitions of limits as x approaches 1, showcasing both valid and invalid cases according to the definition. Core math concepts include limits, continuity, and the epsilon-delta formalism. Constructed using GeoGebra commands like Vector, Segment, Point, and Sequence. Ideal for high school and above students to visually grasp the theory of limits and continuity.

GeoGebra Commands & Parameters

  1. Vector((-0.2, 0), (2.5, 0)) → u
  2. Vector((3.9, 0), (6.5, 0)) → w
  3. Segment((0, 0), (0, 3.5)) → i
  4. Point(i) → A
  5. Vector((0, -1.1), (0, 3.5)) → v
  6. Point(v) → J
  7. Segment(J, K) → j
  8. Segment((0, 0), (2.5, 0)) → k
  9. Vector((4, -1.1), (4, 3.5)) → a
  10. Vector((8, -1.1), (8, 3.5)) → c
  11. Vector((7.9, -1), (10.5, -1)) → b
  12. Sequence(Sequence[Segment[(-0.05 + (4 * l), k), (0.05 + (4 * l), k)], k, -1, 3], l, 0, 1) → VertTickMarks
  13. Point(k) → O
  14. Segment(P, O) → l
  15. Segment(O, (x(O), x(O) + 1)) → m
  16. Segment(P, (x(P), x(P) + 1)) → n
  17. Segment((0, x(P) + 1), (x(P), x(P) + 1)) → t
  18. Segment((0, x(O) + 1), (x(O), x(O) + 1)) → d
  19. Sequence(Sequence[Segment[((4 * l) + k, -0.05), ((4 * l) + k, 0.05)], k, 1, 2], l, 0, 1) → HorizTickMarks
  20. Sequence(Segment[(7.95, k), (8.05, k)], k, 0, 3) → list1
  21. Sequence(Segment[(8 + k, -1.05), (8 + k, -0.95)], k, 1, 2) → list2
  22. Segment((4, -1), (4, 3.5)) → e
  23. Point(e) → B
  24. Point(e) → Q
  25. Segment((4, y(B) - (y(Q) - y(B))), Q) → f_1
  26. Segment((4.1, 0), (5.9, 0)) → g_1
  27. Point(g_1) → R
  28. Segment((5 - (x(R) - 5), 0), R) → h_1
  29. Segment(R, (x(R), x(R) - 5 + 1)) → i_1
  30. Segment((5 - (x(R) - 5), 0), (5 - (x(R) - 5), 2 - (x(R) - 5))) → j_1
  31. Segment((4, x(R) - 5 + 1), (x(R), x(R) - 5 + 1)) → k_1
  32. Segment((4, 1), (5, 1)) → l_1
  33. Segment((4, 2), (5, 2)) → m_1
  34. Segment((4, 5 - (x(R) - 5) - 4 + 1), (5 - (x(R) - 5), 5 - (x(R) - 5) - 4 + 1)) → n_1
  35. Segment((8, -1), (8, 3.5)) → r_1
  36. Point(r_1) → C
  37. Point(r_1) → T
  38. Segment((8, y(C) - (y(T) - y(C))), T) → s_1
  39. Segment((8, -1), (10, -1)) → t_1
  40. Point(t_1) → U
  41. Segment((9 - (x(U) - 9), -1), U) → a_1
  42. Segment(U, (x(U), 3 - (x(U) - 8)^(2))) → b_1
  43. Segment((9 - (x(U) - 9), -1), (18 - x(U), 3 - (18 - x(U) - 8)^(2))) → c_1
  44. Segment((8, 3 - (x(U) - 8)^(2)), (x(U), 3 - (x(U) - 8)^(2))) → d_1
  45. Segment((8, 3 - (18 - x(U) - 8)^(2)), (18 - x(U), 3 - (18 - x(U) - 8)^(2))) → e_1