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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 \((0,0)\).

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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.

What is the limit of (x/y)^3 as (x, y) approaches (0, 0)?

Explore the limit of (x/y)^3 as (x, y) approaches (0, 0). This problem tests key concepts in multivariable calculus, including path dependence in limits. Suitable for advanced high school or college students.

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