The given mathematical expression is:

$\lim_{x \to 0, y \to 0} y \frac{x^2 - y^2}{x^4 + y^2}.$

Step 1: Analyze the behavior of the expression

We need to evaluate the limit of the function as $(x, y) \to (0, 0)$. To determine whether the limit exists, we often analyze the limit along different paths.

Step 2: Substitute different paths

Path 1: $y = 0$

Substitute $y = 0$ into the function: $f(x, y) = y \frac{x^2 - y^2}{x^4 + y^2}.$ If $y = 0$, the numerator becomes: $y (x^2 - 0^2) = 0.$ Thus, along this path: $f(x, 0) = 0.$

Path 2: $x = 0$

Substitute $x = 0$ into the function: $f(x, y) = y \frac{x^2 - y^2}{x^4 + y^2}.$ If $x = 0$, the numerator becomes: $y (-y^2) = -y^3,$ and the denominator becomes: $0^4 + y^2 = y^2.$ Thus, along this path: $f(0, y) = \frac{-y^3}{y^2} = -y.$ As $y \to 0$, this path does not approach a single value.

Step 3: Conclusion

The limit depends on the path taken, as seen above. For example:

• Along $y = 0$, $f(x, y) = 0$.
• Along $x = 0$, $f(x, y) = -y$, which varies based on $y$.

Since the result is path-dependent, the limit does not exist.

Do you want further details or more examples of path evaluation? Here are related questions:

1. What is a step-by-step approach to finding a multivariable limit?
2. How can polar coordinates simplify limit evaluations?
3. What is the significance of path-dependence in multivariable calculus?
4. Can we use the epsilon-delta definition to prove non-existence of limits?
5. How do different coordinate systems (rectangular, polar) aid in analyzing limits?

Tip: When evaluating limits involving two variables, always check multiple paths to test whether the limit depends on direction!