\( \left. \begin{array} { r } { 8 - ) \lim _ { ( x , y ) \rightarrow ( 0,0 ) } \frac { 4 x y } { 3 y ^ { 2 } - x ^ { 2 } } } \\ { x y - 2 x - y + 2 } \end{array} \right. \)

To tackle the limit \( \lim_{(x,y) \rightarrow (0,0)} \frac{4xy}{3y^2 - x^2} \), we can use different paths to investigate if the limit exists. For instance, substituting \( y = kx \) (where \( k \) is a constant) yields different results based on the value of \( k \), suggesting the limit might depend on the path taken toward \( (0,0) \). Now, let’s consider evaluating \( xy - 2x - y + 2 \). By substituting \( (x, y) = (0, 0) \), we get \( 0 - 0 - 0 + 2 = 2 \), which indicates that the expression simplifies nicely to 2 at that point, even though the other limit might not exist. So, double-checking paths and substitutions can unveil interesting insights!