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The problem you provided is the following:

\[
\lim_{(x, y, z) \to (2, 1, 1)} \left( x y^2 - z^2 \right)
\]

We need to compute this limit as \((x, y, z)\) approaches \((2, 1, 1)\).

### Step-by-step solution:

1. **Substitute** \(x = 2\), \(y = 1\), and \(z = 1\) into the given expression \(x y^2 - z^2\).
   
   The expression is:
   \[
   2 \cdot (1)^2 - (1)^2
   \]
   
2. **Simplify** the expression:
   \[
   2 \cdot 1 - 1 = 2 - 1 = 1
   \]

### Final Answer:
\[
\lim_{(x, y, z) \to (2, 1, 1)} \left( x y^2 - z^2 \right) = 1
\]

Would you like further details on this calculation or related topics?

Here are 5 related questions:
1. What are the conditions for the existence of multivariable limits?
2. How can you compute limits if the expression is undefined at the point of interest?
3. What methods can be used to solve limits approaching infinity in multivariable calculus?
4. What is the difference between a partial derivative and a limit in multiple dimensions?
5. How does the concept of continuity extend to functions of several variables?

**Tip:** When evaluating multivariable limits, check if direct substitution works. If not, explore different paths to confirm the limit exists.

Solve the limit problem: \lim_{(x, y, z) \to (2, 1, 1)} \left( x y^2 - z^2 \right)

Learn how to solve the multivariable limit problem \lim_{(x, y, z) \to (2, 1, 1)} \left( x y^2 - z^2 \right). This problem demonstrates the application of direct substitution for limits in multiple dimensions. Suitable for high school or early college-level calculus students.

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