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Example of a Two-Variable Function with No Limit at (0,0)

Example of a Two-Variable Function with No Limit at (0,0)
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2020-02-13 18:20
FunctionCalculusLimitParametric EquationGraphing
① This example demonstrates a two-variable function whose limit at the origin (0,0) does not exist, by showing that different approaching paths yield different limiting values. ② The core mathematical concept is the limit of multivariable functions, emphasizing path-dependence and the criterion for non-existence of a limit. ③ Construction details: The GeoGebra commands used include CurveCartesian to draw space curves and Segment to construct auxiliary segments, with objects such as function f, points A and B, and auxiliary elements a, b, c, d, g, h, e, all visualized in a 3D view to relate the surface and paths. ④ This is suitable for university-level calculus or advanced mathematics courses when teaching multivariable limits, and can be used for classroom demonstrations or student self-exploration to grasp the necessary conditions for a limit to exist.

GeoGebra Commands & Parameters

  1. CurveCartesian(t^(2), t, f(t^(2), t), t, -10, 0) → a
  2. CurveCartesian(t^(2), t, f(t^(2), t), t, 0, 10) → b
  3. CurveCartesian(t^(2), t, t, -10, 10) → c
  4. CurveCartesian(t, 0, f(t, 0), t, -10, 10) → d
  5. CurveCartesian(t, t, t, -10, 10) → g
  6. Segment(B, A) → h
  7. CurveCartesian(t, t, f(t, t), t, -10, 10) → e

AI Drawing Prompt Reference

In a 3D coordinate system, draw the surface of a two-variable function whose limit at the origin does not exist. Include parametric curves representing different approaching paths: one along a parabolic path, one along a straight line in the x-direction, and another along a diagonal line, all extending from negative to positive parameter values and passing through the origin. Also add a straight segment connecting two key points to highlight the differing limit values along different paths. Together, the surface and curves illustrate that as the origin is approached from different directions, the function values tend to different numbers, demonstrating the non-existence of the limit.
The prompt above was compiled from the original drawing commands and is for reference only.