https://file.aiquickdraw.com/mathbot/file/1723261696227yrfqnxnd.jpg

Let's analyze and solve Problem 8 based on the provided image.

### Problem 8
The function \( f(x, y) \) is given as:

\[
f(x, y) = \begin{cases} 
\frac{x^2 \sin \left(\frac{1}{x^2 + y^2}\right)}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

You need to determine which statements are true regarding this function.

#### Step 1: Check for Continuity at \( (0, 0) \)
For \( f(x, y) \) to be differentiable at \( (0, 0) \), it must first be continuous there. We need to check if the limit of \( f(x, y) \) as \( (x, y) \) approaches \( (0, 0) \) is equal to \( f(0, 0) \).

We calculate the limit of \( f(x, y) \) as \( (x, y) \) approaches \( (0, 0) \) along different paths:

- **Path 1:** \( y = 0 \)
\[
f(x, 0) = \frac{x^2 \sin \left(\frac{1}{x^2}\right)}{x^2} = \sin \left(\frac{1}{x^2}\right)
\]
As \( x \to 0 \), \( \sin \left(\frac{1}{x^2}\right) \) oscillates between \(-1\) and \(1\), so the limit does not exist along this path.

- **Path 2:** \( x = 0 \)
\[
f(0, y) = \frac{0 \cdot \sin \left(\frac{1}{y^2}\right)}{y^2} = 0
\]
This path leads to \( 0 \), but the function's behavior is different along different paths, as seen in Path 1.

Because the limit depends on the path taken, \( f(x, y) \) is **not continuous** at \( (0, 0) \).

#### Step 2: Determine Differentiability
Since \( f(x, y) \) is not continuous at \( (0, 0) \), it cannot be differentiable there.

#### Answer for Problem 8:
- (a) **True:** \( f(0, 0) = 0 \)
- (b) **False:** \( f_x(0, 0) \) does not exist because the function is not continuous at \( (0, 0) \).
- (c) **False:** \( f(x, y) \) is not differentiable at \( (0, 0) \).
- (d) **True:** \( f(x, y) \) is not differentiable at \( (0, 0) \).

If you have any questions or need further clarification, feel free to ask!

### Related Questions:
1. What is the difference between partial derivatives and total derivatives?
2. How does the oscillatory behavior of a function affect its continuity?
3. What is the significance of the path taken when evaluating limits in multivariable functions?
4. Can a function have partial derivatives and still be non-differentiable? Why?
5. How do you determine if a function is differentiable at a point?
6. What are common tricks to test for limits in multivariable calculus?
7. How does the sine function's oscillation impact continuity?
8. How does the presence of trigonometric functions in the numerator affect differentiability?

### Tip:
In multivariable calculus, always check the limit along different paths (e.g., \(y = mx\), \(y = x^2\)) when evaluating continuity or differentiability to see if the limit is path-dependent.

Explore the analysis of continuity and differentiability in multivariable functions, focusing on a specific example involving limits and path dependence. Learn about the behavior of functions at critical points and understand the conditions for continuity and differentiability. Suitable for college-level students and enthusiasts of multivariable calculus.

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