We want to show that the limit $\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y^2}$ does not exist. We will do this by approaching $(0,0)$ along two different paths and showing that the limit is different along these paths. The two paths are the x-axis ($y=0$) and the line $y=x$.

1. Problem Description

We want to show that the limit $\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y^2}$ does not exist. We will do this by approaching $(0,0)$ along two different paths and showing that the limit is different along these paths. The two paths are the x-axis ($y=0$) and the line $y=x$.

2. Solution Steps

First, let's consider the path along the x-axis, where $y=0$. Then the expression becomes:

$\lim_{x\to 0} \frac{x(0)}{x^2 + (0)^2} = \lim_{x\to 0} \frac{0}{x^2} = \lim_{x\to 0} 0 = 0$.

So along the x-axis, the limit is

0.

Next, let's consider the path along the line $y=x$. Then the expression becomes:

$\lim_{x\to 0} \frac{x(x)}{x^2 + x^2} = \lim_{x\to 0} \frac{x^2}{2x^2} = \lim_{x\to 0} \frac{1}{2} = \frac{1}{2}$.

So along the line $y=x$, the limit is $\frac{1}{2}$.

Since the limit is 0 along the x-axis and $\frac{1}{2}$ along the line $y=x$, the limit does not exist.

3. Final Answer

The limit does not exist.