Answer
$$ - 1$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,1,1} \right)} \frac{{yz - xy - xz - {x^2}}}{{yz + xy + xz - {y^2}}} \cr
& {\text{Evaluating}} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,1,1} \right)} \frac{{yz - xy - xz - {x^2}}}{{yz + xy + xz - {y^2}}} = \frac{{\left( 1 \right)\left( 1 \right) - \left( 1 \right)\left( 1 \right) - \left( 1 \right)\left( 1 \right) - {{\left( 1 \right)}^2}}}{{\left( 1 \right)\left( 1 \right) + \left( 1 \right)\left( 1 \right) + \left( 1 \right)\left( 1 \right) - {{\left( 1 \right)}^2}}} \cr
& {\text{Simplifying}} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,1,1} \right)} \frac{{yz - xy - xz - {x^2}}}{{yz + xy + xz - {y^2}}} = \frac{{1 - 1 - 1 - 1}}{{1 + 1 + 1 - 1}} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,1,1} \right)} \frac{{yz - xy - xz - {x^2}}}{{yz + xy + xz - {y^2}}} = \frac{{ - 2}}{2} \cr
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {1,1,1} \right)} \frac{{yz - xy - xz - {x^2}}}{{yz + xy + xz - {y^2}}} = - 1 \cr} $$
