Answer
$0$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {2,3} \right)} \frac{{3x - 2y}}{{4{x^2} - {y^2}}} \cr
& {\text{Evaluating the limit by direct substitution}}{\text{.}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {2,3} \right)} \frac{{3x - 2y}}{{4{x^2} - {y^2}}} = \frac{{3\left( 2 \right) - 2\left( 3 \right)}}{{4{{\left( 2 \right)}^2} - {{\left( 3 \right)}^2}}} = \frac{0}{7}=0 \cr
& {\text{Therefore}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {2,3} \right)} \frac{{3x - 2y}}{{4{x^2} - {y^2}}} = 0 \cr} $$
