Why does the multivariable limit of $f(x,y)$=xy/(x²+y²) fail to exist at the origin?

Conditions

• The function is $f(x,y)=xy/(x^2+y^2)$.
• The limit is evaluated as $(x,y) \to (0,0)$.
• The domain excludes the origin $(0,0)$.

Reasoning, step by step

1. State the requirement for a multivariable limit: every domain point sufficiently close to the origin must give an output close to a single value $A$.
2. Evaluate the function along the x-axis ($y=0$): $f(x,0) = 0$, so the limit is 0.
3. Evaluate the function along the y-axis ($x=0$): $f(0,y) = 0$, so the limit is 0.
4. Evaluate the function along the diagonal $y=x$: $f(x,x) = x^2/(2x^2) = 1/2$, so the limit is $1/2$.
5. Evaluate the function along the diagonal $y=-x$: $f(x,-x) = -x^2/(2x^2) = -1/2$, so the limit is -1/2.
6. Compare the results: the outputs along different paths have different limits (0, $1/2$, and -1/2).
7. Conclude that because the limit is not uniform across all paths, the multivariable limit does not exist.

Example

The script states: 'For $f(x,y)$=xy/(x²+y²), coordinate-axis paths give zero, $y=x$ gives $1/2$, and $y=-x$ gives −$1/2$. All approach the origin, but their outputs have different limits. Two differing paths suffice to disprove a limit.'

Common misconceptions

• Believing that agreement along all straight lines is sufficient to prove a limit exists; the video notes that agreement along all straight lines alone need not prove one.
• Thinking that a few plotted paths can prove the existence of a limit; the script clarifies that they 'cannot by themselves prove existence'.
• Confusing one-variable left/right limits with multivariable path limits; in two variables, there are infinitely many paths, not just two sides.

Watch the explanation

Connected concepts

Explore next

• How does the epsilon-delta definition apply to multivariable limits?
• What is the value of $f(x,y)$ along the path y=kx?
• Why is uniform control of every nearby domain point necessary?

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