The given expression is the limit:

$\lim_{(x, y) \to (0,0)} \left( \frac{x}{y} \right)^3$

Let's analyze this limit:

Step-by-Step Solution:

1. Rewrite the limit: The expression inside the limit is $\left( \frac{x}{y} \right)^3$.

2. Consider paths towards $(0, 0)$: To analyze limits in two variables, we check whether the limit depends on the path along which we approach $(0,0)$. If the limit depends on the path, it means the limit does not exist.

   Let's check two specific paths:

   • Path 1: $x = 0$ (approaching along the $y$-axis): $\frac{x}{y} = \frac{0}{y} = 0$ So, the limit along this path becomes: $\lim_{y \to 0} 0^3 = 0$

   • Path 2: $y = x$ (approaching along the line $y = x$): $\frac{x}{y} = \frac{x}{x} = 1$ So, the limit along this path becomes: $\lim_{x \to 0} 1^3 = 1$

3. Conclusion: The limit depends on the path taken. Along the path $x = 0$, the limit is $0$, while along the path $y = x$, the limit is $1$. Since the limit is different for different paths, the limit does not exist.

Final Answer:

The limit does not exist because the value of the expression changes depending on the path taken towards $(0,0)$.

Would you like further details or have any specific questions about this? Here are some related questions for further practice:

1. What happens to the limit if we approach along $y = mx$ (a general line)?
2. How does the limit change if the power is different (e.g., $\left( \frac{x}{y} \right)^n$)?
3. Can polar coordinates help in determining whether a limit exists for two-variable functions?
4. What are some common paths used to check two-variable limits?
5. What are the conditions for a multivariable limit to exist at a point?

Tip: When checking multivariable limits, always test the limit along different paths to ensure consistency. If the result varies with different paths, the limit does not exist.