Let's analyze the given function $g(x, y) = \frac{2x^2y}{x^4 + y^2}$ and evaluate the limit $\lim_{(x, y) \to (0, 0)} g(x, y)$ by testing along various paths.

a. Along the path $x = 0$:

Substitute $x = 0$ into the function: $g(0, y) = \frac{2(0)^2y}{(0)^4 + y^2} = 0$ Thus, the limit along this path is: $\lim_{(x, y) \to (0, 0)} g(x, y) = 0$

b. Along the path $y = 0$:

Substitute $y = 0$ into the function: $g(x, 0) = \frac{2x^2(0)}{x^4 + (0)^2} = 0$ Thus, the limit along this path is: $\lim_{(x, y) \to (0, 0)} g(x, y) = 0$

c. Along the path $y = x^2$:

Substitute $y = x^2$ into the function: $g(x, x^2) = \frac{2x^2(x^2)}{x^4 + (x^2)^2} = \frac{2x^4}{x^4 + x^4} = \frac{2x^4}{2x^4} = 1$ Thus, the limit along this path is: $\lim_{(x, y) \to (0, 0)} g(x, y) = 1$

Conclusion:

The limits are not the same along different paths, so the overall limit does not exist. The limit depends on the path chosen, indicating that $\lim_{(x, y) \to (0, 0)} g(x, y)$ does not exist (DNE).

Would you like more details or have any questions on this solution? Here are some related questions to deepen your understanding:

1. What does it mean for a limit to depend on the path?
2. Can you explain how to determine limits in polar coordinates for multivariable functions?
3. How would the limit change if we evaluated $g(x, y)$ along the path $y = x^n$ for different values of $n$?
4. What are some examples of other functions where the limit at a point does not exist?
5. How can we determine if a limit exists without testing specific paths?

Tip: When testing limits in multivariable calculus, if different paths give different results, the limit does not exist at that point.