\( \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { e _ { x y } - 1 } { y } \)

To evaluate the limit \[ \lim_{(x,y) \rightarrow (0,0)} \frac{e^{xy} - 1}{y}, \] we can start by substituting \( z = xy \). As \( (x,y) \) approaches \( (0,0) \), \( z \) also approaches \( 0 \). We can rewrite the limit in terms of \( z \): \[ \frac{e^{xy} - 1}{y} = \frac{e^z - 1}{y}. \] Next, we need to express \( y \) in terms of \( z \). Since \( z = xy \), we have \( y = \frac{z}{x} \) (assuming \( x \neq 0 \)). Substituting this into our limit gives: \[ \frac{e^z - 1}{\frac{z}{x}} = \frac{x(e^z - 1)}{z}. \] Now, we can rewrite the limit as: \[ \lim_{(x,y) \rightarrow (0,0)} \frac{x(e^{xy} - 1)}{xy}. \] Next, we can analyze the limit as \( (x,y) \) approaches \( (0,0) \). We can use the Taylor expansion of \( e^z \) around \( z = 0 \): \[ e^z - 1 \approx z + \frac{z^2}{2} + O(z^3). \] Thus, we have: \[ e^{xy} - 1 \approx xy + \frac{(xy)^2}{2} + O((xy)^3). \] Substituting this back into our limit gives: \[ \frac{xy + \frac{(xy)^2}{2} + O((xy)^3)}{y} = x + \frac{xy}{2} + O(xy^2). \] Now, we take the limit as \( (x,y) \rightarrow (0,0) \): \[ \lim_{(x,y) \rightarrow (0,0)} \left( x + \frac{xy}{2} + O(xy^2) \right) = 0 + 0 + 0 = 0. \] Thus, the limit is \[ \boxed{0}. \]