Show that

does not exist.

lim (*x*,*y*)→(0,0)sin(xy)/ (*x*+*y*)

To determine whether the limit

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

exists, we can analyze the limit along different paths approaching the origin \((0, 0)\). If the limit is different for two different paths, the limit does not exist.

### Path 1: \( y = x \)
Substitute \( y = x \) into the limit:

\[
\lim_{x \to 0} \frac{\sin(x \cdot x)}{x + x} = \lim_{x \to 0} \frac{\sin(x^2)}{2x}
\]

As \( x \to 0 \), \( x^2 \) also approaches 0. We can use the approximation \( \sin(u) \approx u \) for small \( u \):

\[
\lim_{x \to 0} \frac{x^2}{2x} = \lim_{x \to 0} \frac{x}{2} = 0
\]

So along the path \( y = x \), the limit is 0.

### Path 2: \( y = -x \)
Substitute \( y = -x \) into the limit:

\[
\lim_{x \to 0} \frac{\sin(x \cdot (-x))}{x - x} = \lim_{x \to 0} \frac{\sin(-x^2)}{0}
\]

Here, the expression is undefined because we have division by zero in the denominator.

### Path 3: \( y = 0 \)
Substitute \( y = 0 \) into the limit:

\[
\lim_{x \to 0} \frac{\sin(x \cdot 0)}{x + 0} = \lim_{x \to 0} \frac{0}{x} = 0
\]

So along the path \( y = 0 \), the limit is 0.

### Conclusion
The limit yields different behavior depending on the path taken. For \( y = -x \), the limit is undefined, whereas for other paths, such as \( y = x \) or \( y = 0 \), the limit appears to be 0.

Since the limit depends on the path, **the limit does not exist**.

Would you like any more details or have any further questions?

Here are 5 related questions:
1. How do you formally prove the limit does not exist using the epsilon-delta definition?
2. What happens if you change the function to \( \frac{\sin(xy)}{x^2 + y^2} \)?
3. How can polar coordinates help analyze limits like this?
4. What is an example of a limit that exists even though different paths yield the same value?
5. How does this problem relate to the concept of continuity at a point?

**Tip:** When dealing with limits in multiple variables, checking different paths is a common strategy to see if the limit exists uniformly.

Explore the analysis of the limit \( \lim_{(x, y) \to (0, 0)} \frac{\sin(xy)}{x + y} \), considering different paths approaching the origin. This problem involves techniques from multivariable calculus and limits, assessing the existence of the limit through path analysis.

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