Limits — Question 5

Question 5 – Solution

Strategy. Iterated limits approach along coordinate-aligned stages; the joint limit must agree along every path.

Step 1: Iterated limits For fixed $y\ne 0$, $\lim_{x\to 0}xy/(x^2+y^2)=0$, so $$\lim_{y\to 0}\left(\lim_{x\to 0}f(x,y)\right)=0.$$ Similarly, for fixed $x\ne 0$, the inner $y$-limit is $0$, giving $$\lim_{x\to 0}\left(\lim_{y\to 0}f(x,y)\right)=0.$$

Step 2: Joint paths Along $y=mx$, $$f(x,mx)=\frac{mx^2}{x^2+m^2x^2}=\frac{m}{1+m^2}.$$ This is $0$ for $m=0$ but $1/2$ for $m=1$.

Conclusion. Both iterated limits equal $\boxed{0}$, yet the joint limit $\boxed{\text{does not exist}}$. Iterated limits inspect only two staged approaches and cannot control arbitrary simultaneous approaches.