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Uniqueness of Limits: Theorem and Proof

Uniqueness of Limits: Theorem and Proof
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2018-01-21 07:37
CalculusDerivativeLimitGeometric Construction
This example demonstrates the mathematical theorem and proof process of the Uniqueness of Limits, a fundamental concept in calculus. It utilizes GeoGebra commands like Point, Vector, Textfield, Segment, Line, Derivative, Intersect, Polygon, and Sequence to illustrate the concept, making it suitable for university or advanced high school students studying calculus to visually understand the theorem.

GeoGebra Commands & Parameters

  1. Point(Target) → M1
  2. Point(Target) → LIM
  3. Point(yAxis) → A
  4. Vector(A, LIM) → f_2
  5. Textfield(f) → inputBox1
  6. Vector(x_1, y_1) → v
  7. Segment(f_a, F) → c
  8. Segment(E, F) → d
  9. Segment(A, x_1) → e
  10. Segment(LIM, y_1) → g
  11. Segment(LIM, y_1) → h
  12. Line(x_1, y_1) → i
  13. Derivative(f) → q
  14. Segment(C, D) → b
  15. Segment((0, y(A) - δ), (0, y(A) + δ)) → m
  16. Point(yAxis) → I
  17. Vector(I, J) → u
  18. Intersect(n, p) → K
  19. Intersect(a, p) → L
  20. Intersect(a, r) → M
  21. Intersect(n, r) → N
  22. Polygon(K, L, M, N) → epsdelt_1, k_1, l_1, m_1, n_1
  23. Sequence((0, xk), xk, y(A) - δ, y(A) + δ, δ / 8) → testx
  24. Sequence((2, f(xk)), xk, y(A) - δ, y(A) + δ, δ / 8) → testy
  25. Sequence(Vector[(0, xk), (2, f(xk))], xk, y(A) - δ, y(A) + δ, δ / 8) → list1
  26. Textfield(ε) → inputBox3
  27. Textfield(δ) → inputBox4
  28. Segment(A, G) → l
  29. Sequence(Vector[(0, xk), (2, f(xk))], xk, y(A) + δ / 8, y(A) + δ, δ / 8) → list2
  30. Intersect(a, r) → B
  31. Intersect(a, p) → O
  32. Intersect(p, a_1) → P
  33. Intersect(r, a_1) → Q
  34. Polygon(B, O, P, Q) → epsdelt, b_1, o_1, p_1, q_1
  35. Segment(M10, M11) → f_1