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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)**.

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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.

If g(x, y) = (2x^2y)/(x^4 + y^2), determine lim (x,y)→(0,0) g(x, y) by looking at various paths. Enter DNE if the limit Does Not Exist. Along the paths x = 0, y = 0, and y = x^2.

Explore the process of determining multivariable limits for the function g(x, y) = (2x^2y)/(x^4 + y^2) by analyzing different paths, including x = 0, y = 0, and y = x^2. This problem involves key concepts of multivariable calculus and path-dependent limits, suitable for university-level students.

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