https://file.aiquickdraw.com/mathbot/file/1723046636068fiqux2cx.jpg

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### Problem 8:
Given the function \( f(x, y) \) defined as:
\[ f(x, y) = \begin{cases} 
\frac{7x^2 + y^2}{x^2 + y^2} & \text{if } (x, y) \neq (0,0), \\
0 & \text{if } (x, y) = (0,0),
\end{cases} \]

Options:
a) \( f(x, 0) \) does not exist.
b) \( f(0, y) = 7 \).
c) \( f(0, 0) \) does not exist.
d) \( f(0, 0) = 0 \).

To check the correctness, let's analyze the function.

1. For \( (x, y) \neq (0,0) \):
   \[ f(x, y) = \frac{7x^2 + y^2}{x^2 + y^2} \]
   For specific values:
   - If \( y = 0 \):
     \[ f(x, 0) = \frac{7x^2}{x^2} = 7 \]
   - If \( x = 0 \):
     \[ f(0, y) = \frac{y^2}{y^2} = 1 \]

2. At \( (x, y) = (0,0) \):
   The function is defined as:
   \[ f(0,0) = 0 \]

Now let's examine each option:
- **Option a)** \( f(x, 0) \) does not exist.
  - This is incorrect, \( f(x, 0) = 7 \) for all \( x \neq 0 \).

- **Option b)** \( f(0, y) = 7 \).
  - This is incorrect, \( f(0, y) = 1 \) for all \( y \neq 0 \).

- **Option c)** \( f(0, 0) \) does not exist.
  - This is incorrect, \( f(0, 0) = 0 \).

- **Option d)** \( f(0, 0) = 0 \).
  - This is correct.

Therefore, the correct option for problem 8 is **d)**.

### Problem 9:
Given the function \( f(x, y) \) defined as:
\[ f(x, y) = \begin{cases} 
\frac{x^2 y - x^3 y^2}{x^2 + y^2} & \text{if } (x, y) \neq (0,0), \\
0 & \text{if } (x, y) = (0,0),
\end{cases} \]

Options:
a) \( f(0,0) = 1 \) and \( f(0,0) = 2 \).
b) \( f(0,0) = 0 \) and \( f(x,y) = 0 \).
c) \( f(0,0) = 0 \) and \( f(x,0) = 0 \).
d) \( f(0,0) = 0 \) and \( f(0,y) = 2 \).

To check the correctness, let's analyze the function.

1. For \( (x, y) \neq (0,0) \):
   \[ f(x, y) = \frac{x^2 y - x^3 y^2}{x^2 + y^2} \]
   For specific values:
   - If \( y = 0 \):
     \[ f(x, 0) = \frac{x^2 \cdot 0 - x^3 \cdot 0^2}{x^2 + 0^2} = 0 \]
   - If \( x = 0 \):
     \[ f(0, y) = \frac{0 \cdot y - 0 \cdot y^2}{0 + y^2} = 0 \]

2. At \( (x, y) = (0,0) \):
   The function is defined as:
   \[ f(0,0) = 0 \]

Now let's examine each option:
- **Option a)** \( f(0,0) = 1 \) and \( f(0,0) = 2 \).
  - This is incorrect, \( f(0,0) = 0 \).

- **Option b)** \( f(0,0) = 0 \) and \( f(x,y) = 0 \).
  - This is incorrect, \( f(x,y) \neq 0 \) for all \( (x,y) \neq (0,0) \).

- **Option c)** \( f(0,0) = 0 \) and \( f(x,0) = 0 \).
  - This is correct, since we found \( f(x,0) = 0 \).

- **Option d)** \( f(0,0) = 0 \) and \( f(0,y) = 2 \).
  - This is incorrect, \( f(0,y) = 0 \) for all \( y \neq 0 \).

Therefore, the correct option for problem 9 is **c)**.

### Summary:
- Problem 8: Correct answer is **d)**.
- Problem 9: Correct answer is **c)**.

Would you like more details on these solutions or have any other questions?

Here are some related questions:
1. What is the limit definition of a function's continuity?
2. How can we evaluate the limit of a function with two variables?
3. What is the difference between pointwise and uniform convergence?
4. How can you check if a multivariable function is continuous at a point?
5. What are the different types of discontinuities?
6. How do we differentiate a function of several variables?
7. What are partial derivatives and how are they used?
8. How do we find the gradient of a function with several variables?

**Tip**: Always check the definition of a piecewise function at the boundary points to ensure continuity.

Explore and solve problems involving piecewise functions with detailed solutions. Understand the concept of continuity and limits in piecewise-defined functions. Suitable for high school students studying calculus and advanced algebra.

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