Answer
$$ - 11$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( {3x + 4y - 2} \right) \cr
& {\text{use the law 1 }}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \left( {f\left( {x,y} \right) + g\left( {x,y} \right)} \right).\,\,\,\left( {{\text{see page 887}}} \right){\text{ then}} \cr
& = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( {3x} \right) + \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( {4y} \right) - \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( 2 \right) \cr
& {\text{law constant multiply}} \cr
& = 3\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( x \right) + 4\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( y \right) - \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1, - 3} \right)} \left( 2 \right) \cr
& {\text{evaluate using the theorem 12}}{\text{.1}} \cr
& = 3\left( 1 \right) + 4\left( { - 3} \right) - 2 \cr
& {\text{simplifying}} \cr
& = - 11 \cr} $$
