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This image shows a limit problem involving two variables \( x \) and \( y \). The expression to evaluate is:

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

To determine this limit, we’ll examine the behavior of the expression along different paths approaching \( (0, 0) \). If the limit depends on the path taken, the limit does not exist.

### Step-by-Step Solution

1. **Path \( y = 0 \):**
   \[
   \frac{xy}{x + y} = \frac{x \cdot 0}{x + 0} = 0
   \]

2. **Path \( x = 0 \):**
   \[
   \frac{xy}{x + y} = \frac{0 \cdot y}{0 + y} = 0
   \]

3. **Path \( y = x \):**
   \[
   \frac{xy}{x + y} = \frac{x \cdot x}{x + x} = \frac{x^2}{2x} = \frac{x}{2}
   \]
   As \( x \to 0 \), \( \frac{x}{2} \to 0 \).

4. **Path \( y = -x \):**
   \[
   \frac{xy}{x + y} = \frac{x \cdot (-x)}{x + (-x)} = \frac{-x^2}{0}
   \]
   This results in an undefined expression, suggesting the limit may not exist when approaching along this path.

Since we found an undefined result on the path \( y = -x \), this implies the limit **does not exist**.

Would you like further details on any part of this solution, or have any specific questions?

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Here are five questions related to this concept:

1. How do we determine limits of multivariable functions along different paths?
2. What does it mean if a limit depends on the path taken?
3. Can we use polar coordinates to analyze limits in multivariable functions?
4. What is the significance of an undefined result in evaluating limits?
5. How can the epsilon-delta definition apply to multivariable limits?

**Tip:** In multivariable limits, if different paths yield different results, the limit does not exist.

Evaluate the limit \( \lim_{(x, y) \to (0, 0)} \frac{xy}{x + y} \) using different paths.

Explore how to evaluate the limit \( \lim_{(x, y) \to (0, 0)} \frac{xy}{x + y} \) by examining different paths. This solution demonstrates the use of path dependence in multivariable calculus, including examples of paths where the limit is zero, paths where it is defined, and cases that lead to undefined results.

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