Prove or disprove each of the follow function has limits $x \to a$ by the definition
$\lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2}$
Using $y = mx$ in $\frac{xy}{x^2 + y^2} = \frac{x \cdot mx}{x^2 + (mx)^2} = \frac{m}{1+m^2}$
Meaning it depends on the path, so $\lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2}$ does not exist.
Correct?