Answer
$$6$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,\ln 2,3} \right)} z{e^{xy}} \cr
& {\text{Evaluating}} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,\ln 2,3} \right)} z{e^{xy}} = \left( 3 \right){e^{\left( 1 \right)\left( {\ln 2} \right)}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( 3 \right)\left( 2 \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 6 \cr} $$
