Answer
$$0$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,1,0} \right)} \left( {1 + y} \right)\ln {e^{xz}} \cr
& {\text{Evaluating}} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,1,0} \right)} \left( {1 + y} \right)\ln {e^{xz}} = \left( {1 + 2} \right)\ln {e^{\left( 0 \right)\left( 0 \right)}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( 3 \right)\ln 1 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 0 \cr} $$
