Investigating the Limit of xy/(x²+y²) at (0,0)

Calculus 3 · Axiom Academy

Test the limit along several straight-line paths to decide whether it exists at the origin. Investigate whether the limit exists. If it exists, find its value; if not, explain why. Two directions of approach, two different limiting heights: 0 along the x-axis but ½ along the line y=x . A single limit value cannot be both. Nice work. You showed a two-variable limit does not exist by finding two paths with different limiting values. One path is never enough: for a limit to exist, every path into the point must give the same value. Finding one disagreement settles it: the x-axis gives 0 but the line y=x gives , so no single limit value exists — the limit does not exist. The general line is the clincher: along y=mx the value is , which changes with the slope m (e.g. 0 when m=0 , when m=1 ). Agreeing paths prove nothing: even if many paths had matched, that would only be evidence — never a proof — that a limit exists. Strategy for any two-variable limit: probe a few simple paths first. If two disagree, you are done — the limit does not exist.

This is the written version of the interactive lesson above. See the full Calculus 3 course.