https://file.aiquickdraw.com/mathbot/file/1727063509840su4uih2b.jpg

The image presents three functions of two variables \( f(x, y) \), each defined piecewise. Here's a summary of the three functions:

### 1) First function:
\[
f(x, y) =
\begin{cases} 
\frac{x^3 - y^3}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

### 2) Second function:
\[
f(x, y) =
\begin{cases} 
\frac{2xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

### 3) Third function:
\[
f(x, y) =
\begin{cases} 
\frac{2x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

The functions are discontinuous at the point \( (0, 0) \), and the piecewise definition assigns a value of 0 when \( (x, y) = (0, 0) \).

Would you like to analyze the limits of these functions as \( (x, y) \) approaches \( (0, 0) \), or perhaps explore continuity and differentiability? Let me know how you'd like to proceed!

Here are five related questions you might find interesting:
1. What is the limit of each function as \( (x, y) \to (0, 0) \)?
2. Are any of the given functions continuous at \( (0, 0) \)?
3. What happens if we approach \( (0, 0) \) along different paths?
4. Can these functions be extended to make them continuous at \( (0, 0) \)?
5. What are the partial derivatives of these functions?

**Tip:** When dealing with multivariable limits, it's often useful to approach the point along different paths (e.g., along \( y = 0 \), \( x = 0 \), or \( y = x \)) to check for consistency in the limit value.

Analyze the three given piecewise functions and explore their behavior, continuity, and differentiability at the point (0, 0).

In this problem, we analyze three multivariable piecewise functions, exploring their limits, continuity, and differentiability at the point (0, 0). The functions involve rational expressions and exhibit discontinuities, requiring careful path-based analysis to determine their behavior.

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