	lim┬((x,y)→(0,0) )⁡〖sen(x^2+y^2 )/(3x^2+3y^2 )〗

The given limit is:

\[
\lim_{{(x, y) \to (0, 0)}} \frac{\sin(x^2 + y^2)}{3x^2 + 3y^2}
\]

To solve this limit, let's analyze it using polar coordinates, where \(x = r\cos\theta\) and \(y = r\sin\theta\). In polar coordinates, the expressions \(x^2 + y^2\) and \(3x^2 + 3y^2\) simplify as follows:

\[
x^2 + y^2 = r^2
\]
\[
3x^2 + 3y^2 = 3r^2
\]

Thus, the limit becomes:

\[
\lim_{{r \to 0}} \frac{\sin(r^2)}{3r^2}
\]

Now, consider the small angle approximation for \(\sin(r^2)\) when \(r^2\) is small:

\[
\sin(r^2) \approx r^2
\]

Substituting this approximation into the limit:

\[
\lim_{{r \to 0}} \frac{r^2}{3r^2} = \lim_{{r \to 0}} \frac{1}{3} = \frac{1}{3}
\]

So, the limit is:

\[
\boxed{\frac{1}{3}}
\]

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. How do polar coordinates simplify multivariable limits?
2. What is the small angle approximation and when is it applicable?
3. How can we analyze limits when they approach infinity instead of zero?
4. What are some common pitfalls when evaluating multivariable limits?
5. How does the behavior of sine and cosine functions affect limit evaluations?

**Tip:** Always consider converting to polar coordinates for limits involving \((x, y)\) approaching \((0, 0)\); it often simplifies the expressions.

Learn how to solve the limit of sin(x^2 + y^2) / (3x^2 + 3y^2) as the variables (x, y) approach (0, 0) using polar coordinates and small angle approximation. This problem involves advanced concepts in multivariable calculus and is suitable for college-level mathematics.

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