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

To evaluate the limit \(\lim_{(x,y) \to (0,0)} \frac{x^3 y - xy^3 - 3x^2 - 3y^2}{x^2 + y^2}\), we can use different approaches to determine if the limit exists and, if it does, what the value is.

### 1. **Approach along the x-axis (\(y = 0\)):**

If we set \(y = 0\), the function simplifies to:

\[
f(x, 0) = \frac{x^3(0) - x(0)^3 - 3x^2 - 3(0)^2}{x^2 + 0^2} = \frac{-3x^2}{x^2} = -3
\]

### 2. **Approach along the y-axis (\(x = 0\)):**

If we set \(x = 0\), the function simplifies to:

\[
f(0, y) = \frac{0^3(y) - 0(y)^3 - 3(0)^2 - 3y^2}{0^2 + y^2} = \frac{-3y^2}{y^2} = -3
\]

### 3. **Approach along the line \(y = x\):**

If we set \(y = x\), the function becomes:

\[
f(x, x) = \frac{x^3(x) - x(x)^3 - 3x^2 - 3x^2}{x^2 + x^2} = \frac{x^4 - x^4 - 6x^2}{2x^2} = \frac{-6x^2}{2x^2} = -3
\]

### 4. **Approach along the line \(y = -x\):**

If we set \(y = -x\), the function becomes:

\[
f(x, -x) = \frac{x^3(-x) - x(-x)^3 - 3x^2 - 3(-x)^2}{x^2 + (-x)^2} = \frac{-x^4 - x^4 - 3x^2 - 3x^2}{2x^2} = \frac{-2x^4 - 6x^2}{2x^2} = -3 - x^2
\]

As \(x \to 0\), the \(x^2\) term vanishes, so the limit is still \(-3\).

### Conclusion:
In all these cases, the limit simplifies to \(-3\). Therefore, the limit is:

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

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How can polar coordinates help in evaluating multivariable limits?
2. What are some common methods to determine if a limit does not exist?
3. What are the implications if a limit depends on the path of approach?
4. Can L'Hôpital's Rule be applied to limits in multiple dimensions?
5. What are some challenges when dealing with multivariable limits?

**Tip:** When evaluating multivariable limits, always check the limit along different paths to ensure the limit is consistent or to determine if it does not exist.

Learn how to evaluate the limit \(\lim_{(x,y) \to (0,0)} \frac{x^3 y - xy^3 - 3x^2 - 3y^2}{x^2 + y^2}\) with detailed approaches along different paths. Understand the implications of path dependency in multivariable limits. Suitable for advanced college-level students and anyone studying multivariable calculus.

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