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The given problem is to evaluate the following limit:

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

### Solution:

To determine if the limit exists, we need to check the behavior of the function along different paths approaching the origin \((0, 0)\).

1. **Path 1: Along the x-axis (y = 0)**:
   - Substituting \(y = 0\) into the expression:
     \[
     f(x, 0) = \frac{x(0)}{\sqrt{x^2 + 0^2}} = 0
     \]
   - So, along the x-axis, the limit is \(0\).

2. **Path 2: Along the y-axis (x = 0)**:
   - Substituting \(x = 0\) into the expression:
     \[
     f(0, y) = \frac{0(y)}{\sqrt{0^2 + y^2}} = 0
     \]
   - So, along the y-axis, the limit is also \(0\).

3. **Path 3: Along the line \(y = x\)**:
   - Substituting \(y = x\) into the expression:
     \[
     f(x, x) = \frac{x(x)}{\sqrt{x^2 + x^2}} = \frac{x^2}{\sqrt{2x^2}} = \frac{x^2}{\sqrt{2}|x|} = \frac{|x|}{\sqrt{2}}
     \]
   - As \(x \to 0\), \(\frac{|x|}{\sqrt{2}} \to 0\).
   - Therefore, along this path, the limit is also \(0\).

### Conclusion:
Since the limit is \(0\) along all the paths checked, it seems that the limit exists and equals \(0\). However, the given answer says the limit does not exist. To confirm this, it would be necessary to check more paths or use polar coordinates for a more rigorous approach.

### Polar Coordinates Approach:

In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\), and \(x^2 + y^2 = r^2\). The expression becomes:

\[
f(r, \theta) = \frac{(r \cos \theta)(r \sin \theta)}{\sqrt{r^2}} = \frac{r^2 \cos \theta \sin \theta}{r} = r \cos \theta \sin \theta
\]

As \(r \to 0\), the expression \(r \cos \theta \sin \theta \to 0\) regardless of \(\theta\). Therefore, using polar coordinates also gives a limit of \(0\).

### Final Conclusion:
The limit is \(0\), and it **does exist**. There might be a mistake in the provided answer.

Would you like further details or have any questions?

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Here are 5 related questions:
1. What is a general method for solving two-variable limits?
2. How do polar coordinates help in analyzing two-variable limits?
3. What are some cases where limits do not exist due to different path behaviors?
4. How would the result change if the denominator had a higher power of \(x^2 + y^2\)?
5. What is the significance of using specific paths like \(y = mx\) to evaluate limits?

**Tip**: When checking limits of functions in multiple variables, always try different paths and consider using polar coordinates for a thorough analysis.

Evaluate the limit: lim (x, y) → (0, 0) of (xy) / sqrt(x^2 + y^2).

This problem involves calculating the two-variable limit lim (x, y) → (0, 0) of (xy) / sqrt(x^2 + y^2). The solution explores different approaches, including pathwise analysis and the use of polar coordinates.

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