To solve the limit $\lim_{(x, y) \to (-2, 3)} \frac{x^2}{x^4 + y^2}$, we need to analyze the behavior of the expression as $(x, y)$ approaches $(-2, 3)$.

1. Substitute the values $x = -2$ and $y = 3$ directly into the expression if it results in a defined form:

   $\frac{x^2}{x^4 + y^2} = \frac{(-2)^2}{(-2)^4 + 3^2}$

2. Calculate each part:

   • $x^2 = (-2)^2 = 4$
   • $x^4 = (-2)^4 = 16$
   • $y^2 = 3^2 = 9$

   Substitute these into the expression:

   $\frac{4}{16 + 9} = \frac{4}{25}$

3. Since this results in a specific value without any indeterminate form, the limit exists and is equal to:

   $\frac{4}{25}$

Final Answer:

$\lim_{(x, y) \to (-2, 3)} \frac{x^2}{x^4 + y^2} = \frac{4}{25}$

Would you like further details on any specific step, or have any questions?

Related Questions

1. How would we evaluate a similar limit if it approached a point where the denominator could become zero?
2. What if the numerator had a term involving $y$ as well, like $x^2 + y$?
3. How can polar coordinates be used to evaluate limits of multivariable functions?
4. What does it mean for a limit to be path-dependent in multivariable calculus?
5. How do we determine if a limit does not exist in cases of more complex functions?

Tip:

When evaluating limits in multiple variables, always check if direct substitution yields a finite value before exploring more complex methods like polar coordinates or path analysis.