1. **State the problem:** We need to analyze the function $$f(x,y) = \frac{2x^2y}{x^4 + y^2}$$ and understand its properties. 2. **Examine the domain:** The denominator is $$x^4 + y^2$$ which is always non-negative and zero only at $x=0$ and $y=0$. Therefore, the function is undefined at the origin $(0,0)$. 3. **Analyze behavior near the origin:** To check if the function has a limit or particular behavior near $(0,0)$, try approaching along different paths. - Along $y = 0$, the function becomes $$f(x,0) = \frac{0}{x^4 + 0} = 0.$$ - Along $y = kx^2$ where $k$ is a constant, $$f(x,kx^2) = \frac{2x^2 (k x^2)}{x^4 + (kx^2)^2} = \frac{2 k x^4}{x^4 + k^2 x^4} = \frac{2 k x^4}{x^4 (1 + k^2)} = \frac{2k}{1+k^2}.$$ As $x \to 0$, the expression approaches $\frac{2k}{1+k^2}$, which depends on $k$. 4. **Interpretation:** Since the limit depends on $k$, the limit of $f(x,y)$ as $(x,y) \to (0,0)$ does not exist. 5. **Summary:** The function $$f(x,y) = \frac{2x^2y}{x^4 + y^2}$$ is defined everywhere except at $(0,0)$. The limit at $(0,0)$ does not exist because different paths yield different values. **Final answer:** The function does not have a unique limit at $(0,0)$ and is undefined there.