To show that $\lim_{(x, y) \to (0, 0)} f(x, y) = \frac{xy}{x^2 + y^2}$ does not exist using an $\epsilon$-$\delta$ argument, we demonstrate that the value of the limit depends on the path taken as $(x, y)$ approaches $(0, 0)$. Here's the detailed reasoning:

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Step 1: Path Analysis

The function is given by: $f(x, y) = \frac{xy}{x^2 + y^2}.$

To show that the limit does not exist, we compute the limit along different paths approaching $(0, 0)$ and show that these yield different values.

Path 1: $y = mx$ (a 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 $x^2$ in the denominator: $f(x, mx) = \frac{mx^2}{x^2(1 + m^2)} = \frac{m}{1 + m^2}.$

As $(x, y) \to (0, 0)$, the value of $f(x, y)$ approaches $\frac{m}{1 + m^2}$, which depends on the slope $m$. Thus, the limit depends on the path.

Path 2: $x = 0$ (vertical line)

If $x = 0$, the function becomes: $f(0, y) = \frac{0 \cdot y}{0^2 + y^2} = 0.$

As $(x, y) \to (0, 0)$ along $x = 0$, $f(x, y)$ approaches $0$.

Path 3: $y = 0$ (horizontal line)

If $y = 0$, the function becomes: $f(x, 0) = \frac{x \cdot 0}{x^2 + 0^2} = 0.$

As $(x, y) \to (0, 0)$ along $y = 0$, $f(x, y)$ approaches $0$.

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Step 2: Contradiction in the Limit

From Path 1, the limit depends on $m$, the slope of the line, and can take any value $\frac{m}{1 + m^2}$ depending on $m$. Along the vertical and horizontal lines (Paths 2 and 3), the limit is $0$.

Since the limit depends on the path taken, it is not unique. Thus, the limit does not exist.

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Formal $\epsilon$-$\delta$ Argument

The limit $\lim_{(x, y) \to (0, 0)} f(x, y)$ does not exist because, for any given candidate value $L$, we can find two different paths approaching $(0, 0)$ that lead to different values of $f(x, y)$. This violates the definition of a limit, which requires that all paths approaching the point converge to the same value.

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Summary

The function $\frac{xy}{x^2 + y^2}$ does not have a limit as $(x, y) \to (0, 0)$ because the value of the limit depends on the path taken.

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Would you like further clarification or an illustration of this reasoning?

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Related Questions

1. What does it mean for a limit to exist in terms of paths?
2. Can you provide another example of a function with a non-existent limit?
3. How would you compute the limit if the function had no denominator?
4. What role does the symmetry of $f(x, y)$ play in analyzing the limit?
5. How is this example related to polar coordinate transformations?

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Tip

For multivariable limits, try converting to polar coordinates. This can simplify analysis and help identify dependence on the path.