Evaluate the following limit, if it exists. If the limit does not exist, select 'DNE'.
$$\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$$

• A. DNE
• B. $0$
• C. $\frac{1}{2}$
• D. $1$

Answer: (A)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to evaluate the limit \( \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2} \). This involves determining whether the value of the function approaches a single finite value, diverges, or depends on the path taken as \( (x, y) \to (0, 0) \).
2. Step 2: Analyze the function \( \frac{xy}{x^2 + y^2} \). Notice that the numerator \( xy \) involves the product of \( x \) and \( y \), while the denominator \( x^2 + y^2 \) is the sum of squares, which is always positive except at \( (0, 0) \).
3. Step 3: Test the limit along different paths. For example, substitute \( y = mx \) (a straight line path through the origin) into the function. This gives \( \frac{xy}{x^2 + y^2} = \frac{x(mx)}{x^2 + (mx)^2} = \frac{m x^2}{x^2 + m^2 x^2} = \frac{m}{1 + m^2} \). The result depends on \( m \), indicating the limit may depend on the path.
4. Step 4: Test another path, such as \( y = 0 \). Substituting \( y = 0 \) into the function gives \( \frac{xy}{x^2 + y^2} = \frac{x(0)}{x^2 + 0^2} = 0 \). This result is different from the previous path, further suggesting the limit depends on the path.
5. Step 5: Conclude that the limit does not exist (DNE). Since the value of \( \frac{xy}{x^2 + y^2} \) depends on the path taken as \( (x, y) \to (0, 0) \), the limit does not approach a single finite value. Therefore, the correct answer is DNE.