Show that
lim
(x,y)→(0,0)
∴      lim
f (x, y) = 0.
■
xy
x  −y
22
x  +y
2         2  = 0

It seems like you are asking to evaluate the limit of a function \( f(x, y) \) as \( (x, y) \to (0, 0) \). However, the expression appears unclear due to formatting issues. Let's carefully break it down and understand what the function might be before we proceed with the evaluation.

Based on your input, it seems like the function you're referring to is:

\[
f(x, y) = \frac{xy}{x^2 + y^2}
\]

You want to show that:

\[
\lim_{(x, y) \to (0, 0)} f(x, y) = 0
\]

### Step 1: Check along the x-axis (\(y = 0\))
Substitute \( y = 0 \) into the function:

\[
f(x, 0) = \frac{x \cdot 0}{x^2 + 0^2} = 0
\]

So, along the x-axis, the function \( f(x, y) \) equals 0.

### Step 2: Check along the y-axis (\(x = 0\))
Substitute \( x = 0 \) into the function:

\[
f(0, y) = \frac{0 \cdot y}{0^2 + y^2} = 0
\]

So, along the y-axis, the function \( f(x, y) \) also equals 0.

### Step 3: Check along the line \( y = x \)
Substitute \( y = x \) into the function:

\[
f(x, x) = \frac{x \cdot x}{x^2 + x^2} = \frac{x^2}{2x^2} = \frac{1}{2}
\]

Along the line \( y = x \), the function approaches \( \frac{1}{2} \), which is not equal to 0.

### Conclusion:
Since the limit depends on the path of approach, the limit as \( (x, y) \to (0, 0) \) does not exist. Along some paths, the value is 0, while along others (such as \( y = x \)), the value is not 0.

Thus, the limit does **not exist**.

Would you like further details or have any questions?

Here are some related questions to consider:
1. What happens if you approach the origin along a parabolic path, like \( y = x^2 \)?
2. How can polar coordinates help evaluate limits involving two variables?
3. Can you think of a case where a limit depends on direction, but still exists?
4. What other strategies can be used to confirm the non-existence of a limit?
5. How does the epsilon-delta definition of limits work in multivariable calculus?

**Tip:** When dealing with limits in multiple variables, always check different paths and consider using polar coordinates to check for consistency.

Show that lim (x,y)→(0,0) f(x, y) = 0 for f(x, y) = xy / (x^2 + y^2).

Explore how to evaluate the multivariable limit lim (x, y)→(0, 0) of the function f(x, y) = xy / (x^2 + y^2). This problem examines the path dependency in multivariable calculus, revealing how different approaches yield different results. Suitable for college-level calculus students.

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