The given function is:

$f(x, y) = -\frac{xy}{x^2 + y^2}.$

The task is to determine the limit of $f(x, y)$ as $(x, y) \to (0, 0)$ along specific paths.

General Approach:

To find the limit along different paths, substitute specific relationships between $x$ and $y$ into the function and evaluate the behavior as $(x, y) \to (0, 0)$.

Path 1: Along $y = mx$ (straight line through the origin)

Substitute $y = mx$ into $f(x, y)$: $f(x, mx) = -\frac{x(mx)}{x^2 + (mx)^2} = -\frac{mx^2}{x^2 + m^2x^2}.$ Factor out $x^2$ from the denominator: $f(x, mx) = -\frac{mx^2}{x^2(1 + m^2)} = -\frac{m}{1 + m^2}.$ This is a constant and does not depend on $x$. Thus, along $y = mx$, the limit is: $\lim_{(x, y) \to (0, 0)} f(x, y) = -\frac{m}{1 + m^2}.$

Path 2: Along $y = 0$ (x-axis)

Substitute $y = 0$ into $f(x, y)$: $f(x, 0) = -\frac{x(0)}{x^2 + 0^2} = 0.$ Thus, along the x-axis, the limit is: $\lim_{(x, y) \to (0, 0)} f(x, y) = 0.$

Path 3: Along $x = 0$ (y-axis)

Substitute $x = 0$ into $f(x, y)$: $f(0, y) = -\frac{(0)y}{0^2 + y^2} = 0.$ Thus, along the y-axis, the limit is: $\lim_{(x, y) \to (0, 0)} f(x, y) = 0.$

Path 4: Along $y = x^2$ (parabolic path)

Substitute $y = x^2$ into $f(x, y)$: $f(x, x^2) = -\frac{x(x^2)}{x^2 + (x^2)^2} = -\frac{x^3}{x^2 + x^4}.$ Factor $x^2$ from the denominator: f(x, x^2) = -\frac{x^3}{x^2(1 + x^2)} = -\frac{x}{1 + x^2}$$.

As $x \to 0$, the numerator $x$ approaches 0, and the denominator $1 + x^2$ approaches 1. Thus, along $y = x^2$, the limit is: [ \lim_{(x, y) \to (0, 0)} f(x, y) = 0.$$

Conclusion:

The limit depends on the path:

1. Along $y = mx$, the limit is $-\frac{m}{1 + m^2}$, which varies with $m$.
2. Along $y = 0$ or $x = 0$, or along $y = x^2$, the limit is $0$.

Since the limit depends on the path, the two-dimensional limit $\lim_{(x, y) \to (0, 0)} f(x, y)$ does not exist.

Would you like more clarification or details about these calculations? Here are 5 related questions you might explore:

1. How do we formally prove that a multivariable limit does not exist?
2. Can this function be made continuous at $(0, 0)$ by redefining $f(0, 0)$?
3. How does the behavior of the function change for other parabolic paths like $y = ax^2$?
4. What does the gradient of $f(x, y)$ at points away from the origin suggest about the behavior near $(0, 0)$?
5. How can polar coordinates be used to evaluate this limit?

Tip: When working with multivariable limits, always test different paths to check if the limit depends on the direction of approach.