We are tasked with finding the limit of the following expression as $(x, y) \to (0, 0)$:

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

Step 1: Rewrite the expression

Let’s first observe the structure of the problem. The numerator involves $\sin^2(x^2 + y^2)$, and the denominator involves $x^4 + y^4$. It is helpful to rewrite the expression in polar coordinates to exploit the symmetry of the problem.

In polar coordinates:

• $x = r \cos(\theta)$
• $y = r \sin(\theta)$
• $x^2 + y^2 = r^2$

Thus, the expression becomes: $\frac{\sin^2(r^2)}{x^4 + y^4}$

Now, for $x^4 + y^4$, we can express it as: $x^4 + y^4 = r^4 \left( \cos^4(\theta) + \sin^4(\theta) \right)$

Therefore, the limit expression becomes: $\frac{\sin^2(r^2)}{r^4 \left( \cos^4(\theta) + \sin^4(\theta) \right)}$

Step 2: Analyze the behavior as $r \to 0$

We now analyze the behavior of the expression as $r \to 0$.

• The term $\sin^2(r^2)$ behaves like $r^4$ for small $r$, since $\sin(z) \approx z$ for small $z$. Therefore, $\sin^2(r^2) \approx r^4$.
• Thus, the expression simplifies to: $\frac{r^4}{r^4 \left( \cos^4(\theta) + \sin^4(\theta) \right)} = \frac{1}{\cos^4(\theta) + \sin^4(\theta)}$

Step 3: Conclusion

As $r \to 0$, the expression tends to: $\frac{1}{\cos^4(\theta) + \sin^4(\theta)}$

This shows that the limit depends on $\theta$, which means the limit does not exist in the conventional sense. The value of the limit varies depending on the direction from which we approach $(0, 0)$. Thus, the limit does not exist.

Would you like further details or clarification?

Here are some related questions:

1. How can we determine whether a limit exists for functions of two variables?
2. How do we analyze limits of multi-variable functions using polar coordinates?
3. Can you describe the behavior of $\sin(x^2 + y^2)$ for small values of $x$ and $y$?
4. What happens if we change the denominator to $x^2 + y^2$ instead of $x^4 + y^4$?
5. How can we use path-dependent limits to test for the existence of a limit in multi-variable functions?

Tip: In multi-variable limits, if the limit depends on the path you take to approach the point, the limit does not exist.