To evaluate the limit

$\lim_{(x, y) \to (0, 0)} \frac{y^2 \sin^2 x}{x^4 + y^4}$

we'll analyze how the expression behaves as $(x, y)$ approaches $(0, 0)$ from different paths. Often, using polar coordinates or testing specific paths can help reveal if the limit exists.

Step-by-Step Solution

1. Convert to Polar Coordinates: In polar coordinates, $x = r \cos \theta$ and $y = r \sin \theta$. Then, $x^2 + y^2 = r^2$, and the function transforms accordingly: $y^2 = r^2 \sin^2 \theta \quad \text{and} \quad x^4 + y^4 = r^4 (\cos^4 \theta + \sin^4 \theta).$

   Substitute into the expression: $\frac{y^2 \sin^2 x}{x^4 + y^4} = \frac{r^2 \sin^2 \theta \sin^2 (r \cos \theta)}{r^4 (\cos^4 \theta + \sin^4 \theta)}.$

2. Simplify the Expression: Simplify by canceling $r^2$ from the numerator and denominator: $= \frac{\sin^2 \theta \sin^2 (r \cos \theta)}{r^2 (\cos^4 \theta + \sin^4 \theta)}.$

3. Analyze the Behavior of $\sin^2 (r \cos \theta)$ as $r \to 0$: Notice that $\sin^2 (r \cos \theta)$ behaves approximately like $(r \cos \theta)^2$ for small $r$. Thus, $\sin^2 (r \cos \theta) \approx r^2 \cos^2 \theta.$

   Substitute this approximation: $\frac{\sin^2 \theta \cdot r^2 \cos^2 \theta}{r^2 (\cos^4 \theta + \sin^4 \theta)} = \frac{r^2 \sin^2 \theta \cos^2 \theta}{r^2 (\cos^4 \theta + \sin^4 \theta)}.$

   Cancel $r^2$: $= \frac{\sin^2 \theta \cos^2 \theta}{\cos^4 \theta + \sin^4 \theta}.$

4. Check if the Result Depends on $\theta$: This expression depends on $\theta$ because $\sin^2 \theta \cos^2 \theta$ and $\cos^4 \theta + \sin^4 \theta$ vary with $\theta$. Therefore, the limit depends on the path taken to approach $(0,0)$, which means the limit does not exist.

Conclusion

Since the limit depends on the angle $\theta$, it does not converge to a single value as $(x, y) \to (0,0)$. Thus:

$\lim_{(x, y) \to (0, 0)} \frac{y^2 \sin^2 x}{x^4 + y^4} \quad \text{does not exist}.$

Would you like further details or have questions about any specific steps?

Here are some related questions to further explore this topic:

1. What is the importance of converting to polar coordinates in multivariable limits?
2. How can testing different paths help in determining the existence of a limit?
3. Can we use other coordinate transformations, like spherical coordinates, for limits in higher dimensions?
4. What are some common cases where limits do not exist due to path dependence?
5. How does L'Hospital's Rule extend (if at all) to multivariable limits?

Tip: When checking multivariable limits, try substituting polar coordinates or testing specific paths. It’s often helpful in determining if a limit exists or depends on the approach path.