https://file.aiquickdraw.com/mathbot/file/1726615708179a83pm6i8.jpg

The image contains four limits (problems 8, 9, 10, and 11). I'll walk through each one step-by-step.

### Problem 8:
\[
\lim_{(x, y) \to (0, 0)} \frac{1 - \cos(x^2 + y^2)}{2(x^2 + y^2)}
\]

#### Solution:
1. Let \( r^2 = x^2 + y^2 \), which simplifies the expression.
   \[
   \lim_{r \to 0} \frac{1 - \cos(r^2)}{2r^2}
   \]

2. Using the approximation for small values \( \cos(r^2) \approx 1 - \frac{r^4}{2} \), we substitute into the limit:
   \[
   \frac{1 - (1 - \frac{r^4}{2})}{2r^2} = \frac{\frac{r^4}{2}}{2r^2} = \frac{r^2}{4}
   \]

3. As \( r \to 0 \), the expression \( \frac{r^2}{4} \to 0 \).

Thus, the limit is:
\[
\boxed{0}
\]

---

### Problem 9:
\[
\lim_{(x, y) \to (0, 0)} \frac{x^4 - 16y^4}{x^2 - 4y^2}
\]

#### Solution:
1. Factor the numerator and denominator:
   \[
   \frac{(x^2 - 4y^2)(x^2 + 4y^2)}{(x^2 - 4y^2)}
   \]

2. Cancel the common factor \( (x^2 - 4y^2) \) (for \( x^2 - 4y^2 \neq 0 \)):
   \[
   x^2 + 4y^2
   \]

3. Now, evaluate the limit:
   \[
   \lim_{(x, y) \to (0, 0)} (x^2 + 4y^2) = 0
   \]

Thus, the limit is:
\[
\boxed{0}
\]

---

### Problem 10:
\[
\lim_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2 + y^2}}
\]

#### Solution:
1. Use polar coordinates where \( x = r\cos\theta \) and \( y = r\sin\theta \), and \( r^2 = x^2 + y^2 \).
   The expression becomes:
   \[
   \frac{r\cos\theta r\sin\theta}{r} = r\cos\theta \sin\theta
   \]

2. As \( r \to 0 \), \( r\cos\theta \sin\theta \to 0 \), regardless of \( \theta \).

Thus, the limit is:
\[
\boxed{0}
\]

---

### Problem 11:
\[
\lim_{(x, y) \to (0, 0)} \frac{x^3 y^3}{x^2 + y^2}
\]

#### Solution:
1. Again, use polar coordinates, \( x = r\cos\theta \), \( y = r\sin\theta \), and \( r^2 = x^2 + y^2 \). The expression becomes:
   \[
   \frac{(r\cos\theta)^3 (r\sin\theta)^3}{r^2} = \frac{r^6\cos^3\theta\sin^3\theta}{r^2} = r^4 \cos^3\theta \sin^3\theta
   \]

2. As \( r \to 0 \), \( r^4 \to 0 \).

Thus, the limit is:
\[
\boxed{0}
\]

---

Let me know if you'd like further details or explanations!

#### Five related questions:
1. How are polar coordinates useful in multivariable calculus?
2. What are common limit techniques used to evaluate two-variable limits?
3. What is the significance of the squeeze theorem in limit calculations?
4. How does L'Hopital's Rule apply to limits with indeterminate forms?
5. How can trigonometric approximations help simplify complex limits?

#### Tip:
When evaluating limits at \( (0, 0) \), always check if polar coordinates can simplify the expression by combining terms involving \( x^2 + y^2 \) into a single variable \( r \).

Solve the limits for problems 8, 9, 10, and 11 in the image.

This page provides step-by-step solutions to four multivariable limit problems involving polar coordinates and indeterminate forms. It covers essential techniques like factoring, trigonometric approximations, and polar transformations to simplify complex limits.

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