Let's evaluate the limit \( \lim_{(x, y) \to (0, 0)} 3xy \) and determine whether it equals 6.

### Step-by-Step Solution:

The given expression is \( 3xy \), and we want to find:

\[
\lim_{(x, y) \to (0, 0)} 3xy
\]

#### 1. Substituting \( (x, y) = (0, 0) \):
At \( x = 0 \) and \( y = 0 \), the function \( 3xy \) becomes:

\[
3 \cdot 0 \cdot 0 = 0
\]

So, the function tends to 0, not 6, as \( (x, y) \to (0, 0) \).

#### 2. Verifying the limit along different paths:
To confirm that the limit is 0, let's check the behavior of the function along different paths toward \( (0, 0) \).

- **Path 1**: Along \( y = 0 \), we get:
  \[
  3xy = 3x \cdot 0 = 0
  \]
  So, along this path, \( \lim_{x \to 0} 3xy = 0 \).

- **Path 2**: Along \( x = 0 \), we get:
  \[
  3xy = 3 \cdot 0 \cdot y = 0
  \]
  So, along this path, \( \lim_{y \to 0} 3xy = 0 \).

- **Path 3**: Along \( y = x \), we get:
  \[
  3xy = 3x \cdot x = 3x^2
  \]
  Now, taking the limit as \( x \to 0 \):
  \[
  \lim_{x \to 0} 3x^2 = 0
  \]

Since the function tends to 0 along multiple paths, we conclude that:

\[
\lim_{(x, y) \to (0, 0)} 3xy = 0
\]

### Conclusion:
The limit \( \lim_{(x, y) \to (0, 0)} 3xy \) does not equal 6. In fact, it equals 0.

---

Would you like more details or clarification? Here are five related questions for practice:

1. What is \( \lim_{(x, y) \to (0, 0)} xy \)?
2. How do you evaluate \( \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2} \)?
3. Can a multivariable limit exist if the value differs along different paths?
4. What is the geometric interpretation of limits in two variables?
5. How can polar coordinates help in evaluating multivariable limits?

**Tip:** When evaluating limits in two variables, always check the behavior along different paths to ensure consistency!

Discover the step-by-step process for proving that the multivariable limit lim (x, y) → (0, 0) 3xy does not equal 6. This solution uses paths like y = 0 and x = y to show that the limit is 0. Ideal for undergraduate students studying multivariable calculus.

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