The given limit is:

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

### Solution:

We'll approach this limit using polar coordinates to simplify the expression. In polar coordinates, we make the substitutions:

- \( x = r\cos\theta \)
- \( y = r\sin\theta \)
- \( x^2 + y^2 = r^2 \)

Thus, the expression becomes:

\[
\frac{xy}{x^2 + y^2} = \frac{(r\cos\theta)(r\sin\theta)}{r^2} = \frac{r^2\cos\theta\sin\theta}{r^2}
\]

This simplifies to:

\[
\cos\theta \sin\theta
\]

Now, as \( r \to 0 \), the function \( \cos\theta \sin\theta \) depends only on \( \theta \), but it is bounded between -1 and 1. The function does not approach a single value as \( r \to 0 \). Hence, the limit **does not exist**.

### Conclusion:

\[
\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2} \text{ does not exist.}
\]

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Do you want further details or have any questions?

Here are 5 related questions for further practice:
1. Evaluate \( \lim_{(x,y) \to (0,0)} \frac{x^2 - y^2}{x^2 + y^2} \).
2. Check if the limit \( \lim_{(x,y) \to (0,0)} \frac{x^3 + y^3}{x^2 + y^2} \) exists.
3. Explore the behavior of \( \lim_{(x,y) \to (0,0)} \frac{x^2y}{x^2 + y^2} \).
4. Calculate \( \lim_{(x,y) \to (0,0)} \frac{xy^2}{x^2 + y^4} \).
5. Analyze the limit \( \lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^4} \).

**Tip**: When dealing with two-variable limits, polar coordinates are often useful for simplifying expressions and analyzing the behavior as variables approach zero.

Explore the solution for lim(x,y)→(0,0) xy / (x^2 + y^2), showing why the limit does not exist using polar coordinates. The solution involves key concepts in multivariable calculus, including polar coordinate substitution and limit analysis. Suitable for university-level calculus students.

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