Answer
$$2$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( {10x - 5y + 6xy} \right) \cr
& {\text{use the law 1 }}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \left( {f\left( {x,y} \right) + g\left( {x,y} \right)} \right).\,\,\,\left( {{\text{see page 887}}} \right){\text{then}} \cr
& = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( {10x} \right) - \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( {5y} \right) + \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( {6xy} \right) \cr
& {\text{law constant multiply}} \cr
& = 10\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( x \right) - 5\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( y \right) + 6\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {4, - 2} \right)} \left( {xy} \right) \cr
& {\text{evaluate using the theorem 12}}{\text{.1}} \cr
& = 10\left( 4 \right) - 5\left( { - 2} \right) + 6\left( 4 \right)\left( { - 2} \right) \cr
& {\text{simplifying}} \cr
& = 40 + 10 - 48 \cr
& = 2 \cr} $$
