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Formal Limit Definition Demonstration

Formal Limit Definition Demonstration
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2016-01-14 06:17
FunctionCalculusLimitGeometric Construction
In collections Fun with Functions
This example illustrates the formal definition of limits. The student can set an epsilon value and then decrease delta until the range is less than epsilon, visually understanding the concept of limits. The core math concept involved is the theory of limits. Construction highlights include using GeoGebra commands and objects such as Point, Element, If, Max, Textfield, OrthogonalLine, Polygon. Suitable for high school and above students learning calculus.

GeoGebra Commands & Parameters

  1. Point(xAxis) → goal
  2. Point(xAxis) → SplitPoint
  3. Element(flist, nf) → g
  4. If((xm ≤ 1) ∧ (xp ≥ 1), Infinity, Min[f_7(xm), f_7(xp)]) → ym7
  5. If(xm ≥ xSplit, y(Min[f_{right}, xm, xp]), If[xp < xSplit, y(Min[f_{left}, xm, xp]), Min[y(Min[f_{left}, xm, xSplit]), y(Min[f_{right}, xSplit, xp])]]) → ym5
  6. If((xp * xm) ≤ 0, 0, Min[If[xp < 0, (-xp), If[xp ≤ 1, xp^(2), 1]], If[xm < 0, (-xm), If[xm ≤ 1, xm^(2), 1]]]) → ym3
  7. If((xp ≥ 1) ∧ (xm ≤ 1) ∧ (aa ≠ 1), 2, xp) → yx8
  8. If((xm ≤ 1) ∧ (xp ≥ 1), Infinity, Max[f_7(xm), f_7(xp)]) → yx7
  9. If(xm ≥ xSplit, y(Max[f_{right}, xm, xp]), If[xp < xSplit, y(Max[f_{left}, xm, xp]), Max[y(Max[f_{left}, xm, xSplit]), If[xp ≟ xSplit, f_{right}(xSplit), y(Max[f_{right}, xSplit, xp])]]]) → yx5
  10. Max(If[xm < 0, (-xm), If[xm ≤ 1, xm^(2), 1]], If[xp < 0, 0, If[xp ≤ 1, xp^(2), 1]]) → yx3
  11. If(nf < 8, g(aa), If[nf ≟ 8 ∧ aa ≟ 1, 1, If[nf ≟ 9 ∧ aa ≟ 0, 2.7183, g(aa)]]) → L
  12. Element(maxlist, nf) → ymax
  13. Element(minlist, nf) → ymin
  14. Max(L - ymin, ymax - L) → epsilon
  15. Textfield(f_{left}) → inputBox1
  16. Textfield(f_{right}) → inputBox2
  17. OrthogonalLine(F, xAxis) → a
  18. OrthogonalLine(Xm, xAxis) → b
  19. OrthogonalLine(Xp, xAxis) → c
  20. OrthogonalLine(Ymin, yAxis) → d
  21. OrthogonalLine(Ymax, yAxis) → e
  22. Textfield(egoal) → inputBox3
  23. Polygon(G, C, B, A) → poly1, g_1, c_1, b_1, a_1

AI Drawing Prompt Reference

Draw a function graph to demonstrate the formal definition of limits. Set an epsilon value and gradually decrease delta until the range is less than epsilon, visually understanding the concept of limits. Include points on the x-axis and function value points, connect key points with orthogonal lines, display the epsilon and delta ranges, and highlight important areas with a polygon.
The prompt above was compiled from the original drawing commands and is for reference only.