The given limit problem is:

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

Step 1: Analyzing the Limit

We need to evaluate the limit as $(x, y)$ approaches $(0, 0)$. This is a multivariable limit, so the approach to zero can vary based on the path we choose. To verify whether the limit exists, we can evaluate the limit along different paths.

Step 2: Path 1: Along $y = 0$

When $y = 0$, the expression becomes:

$\frac{x^3(0) - 2(0)}{x^2 + x(0) + (0)^2} = \frac{0}{x^2} = 0$

So, along $y = 0$, the limit is 0.

Step 3: Path 2: Along $x = 0$

When $x = 0$, the expression becomes:

$\frac{0^3(y) - 2y}{0^2 + 0(y) + y^2} = \frac{-2y}{y^2} = \frac{-2}{y}$

As $y \to 0$, $\frac{-2}{y}$ does not approach a finite value; instead, it goes to infinity or negative infinity depending on the direction from which $y$ approaches 0. Therefore, the limit does not exist along this path.

Conclusion:

Since the limit behaves differently along different paths, the overall limit does not exist.

Would you like more details or have any questions about the steps?

Here are 5 related questions to further your understanding:

1. How do you determine if a multivariable limit exists?
2. Why is evaluating limits along different paths important for multivariable limits?
3. What does it mean if a limit approaches different values along different paths?
4. Can the polar coordinate substitution method help for other multivariable limit problems?
5. What is the definition of continuity for a function of multiple variables?

Tip: When dealing with multivariable limits, always try evaluating the function along different paths to check if the limit is path-dependent.