1. **State the problem:** We want to show that the limit $$\lim_{(x,y) \to (0,0)} \frac{xy}{x^4 + y^2}$$ does not exist by approaching along different paths, including the path $$y = kx^2$$. 2. **Approach along the path $$y=0$$:** Substitute $$y=0$$ into the function: $$f(x,0) = \frac{x \cdot 0}{x^4 + 0^2} = 0$$ As $$x \to 0$$, $$f(x,0) \to 0$$. 3. **Approach along the path $$x=0$$:** Substitute $$x=0$$: $$f(0,y) = \frac{0 \cdot y}{0^4 + y^2} = 0$$ As $$y \to 0$$, $$f(0,y) \to 0$$. 4. **Approach along the path $$y = kx^2$$:** Substitute $$y = kx^2$$: $$f(x,kx^2) = \frac{x \cdot kx^2}{x^4 + (kx^2)^2} = \frac{kx^3}{x^4 + k^2 x^4} = \frac{kx^3}{x^4(1 + k^2)} = \frac{k}{1 + k^2} \cdot \frac{1}{x}$$ 5. **Analyze the limit along $$y = kx^2$$:** As $$x \to 0$$, $$\frac{1}{x}$$ becomes unbounded (tends to $$\pm \infty$$ depending on the direction). Therefore, the limit does not approach a finite value along this path. 6. **Conclusion:** Since the limit along $$y=0$$ and $$x=0$$ is 0, but along $$y = kx^2$$ the function becomes unbounded and does not approach a finite limit, the overall limit $$\lim_{(x,y) \to (0,0)} \frac{xy}{x^4 + y^2}$$ does not exist.