Functions of Several Variables — Question 6

Question 6

Investigate $$\lim_{(x,y)\to(0,0)}\frac{x^2-y^2}{x^2+y^2}.$$ Evaluate along at least three families of paths and explain why the evidence is decisive.

Question 6 – Solution

Strategy A multivariable limit must have the same value along every path.

Path tests Along $y=0$, the expression equals $1$; along $x=0$, it equals $-1$; and along $y=mx$, it equals $$\frac{1-m^2}{1+m^2},$$ which varies continuously with $m$.

Conclusion Since even two paths give different values, the limit $\boxed{\text{does not exist}}$.

Why decisive Path agreement can only suggest a limit, but one disagreement proves nonexistence because the definition requires uniform behavior over all approaches.