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The problem asks to analyze the existence of the following limit, and if it doesn't exist, to justify it, otherwise, to find its value:

\[
\lim_{(x,y) \to (0,1)} \frac{x(y - 1)^3 + (y - 1)^6}{3x^2 + (y - 1)^6}
\]

### Step-by-step analysis:

1. **Substitute the point** \((x, y) = (0,1)\):
   - When \(x = 0\) and \(y = 1\), the numerator becomes:
     \[
     0 \cdot (1-1)^3 + (1-1)^6 = 0
     \]
   - The denominator becomes:
     \[
     3 \cdot 0^2 + (1-1)^6 = 0
     \]
   - Hence, the expression takes the indeterminate form \(\frac{0}{0}\).

2. **Approach along different paths:**

   To determine whether the limit exists, we can evaluate the limit along different paths and check if the value is consistent.

   - **Path 1: Along the line \(x = 0\):**
     When \(x = 0\), the expression reduces to:
     \[
     \frac{(y - 1)^6}{(y - 1)^6} = 1
     \]
     So, the limit along this path is 1.

   - **Path 2: Along the line \(y = 1\):**
     When \(y = 1\), the expression reduces to:
     \[
     \frac{0 \cdot (1-1)^3 + 0}{3x^2 + 0} = 0
     \]
     So, the limit along this path is 0.

Since the limit gives different results along different paths (1 and 0), **the limit does not exist**.

### Conclusion:
The limit does not exist because the value depends on the path taken.

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Do you want further details on this solution, or have any questions? Here are some related questions for further exploration:

1. What other paths could you take to evaluate a two-variable limit?
2. How can polar coordinates be used to analyze two-variable limits?
3. What is the formal definition of a two-variable limit?
4. Can you give an example of a limit that does exist and explain why?
5. How do indeterminate forms affect the computation of limits in multivariable calculus?

**Tip:** When a limit appears in the indeterminate form \(\frac{0}{0}\), always consider testing multiple paths to determine if the limit exists.

Analice la existencia del siguiente límite: lim (x,y)→(0,1) (x(y−1)^3 + (y−1)^6) / (3x^2 + (y−1)^6)

Explore the detailed solution for the two-variable limit problem lim (x,y)→(0,1) (x(y−1)^3 + (y−1)^6) / (3x^2 + (y−1)^6). The solution includes path-based evaluation techniques and discusses why the limit does not exist due to different results along different paths.

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