Answer
$$1$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{{x^2} + xy - 2{y^2}}}{{2{x^2} - xy - {y^2}}} \cr
& {\text{Evaluating the limit}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{{x^2} + xy - 2{y^2}}}{{2{x^2} - xy - {y^2}}} = \frac{{{{\left( 1 \right)}^2} + \left( 1 \right)\left( 1 \right) - 2{{\left( 1 \right)}^2}}}{{2{{\left( 1 \right)}^2} - \left( 1 \right)\left( 1 \right) - {{\left( 1 \right)}^2}}} = \frac{0}{0} \cr
& {\text{Factor the numerator and denominator}} \cr
& {x^2} + xy - 2{y^2} = \left( {x + 2y} \right)\left( {x - y} \right) \cr
& 2{x^2} - xy - {y^2} = \left( {2x + y} \right)\left( {x - y} \right) \cr
& {\text{Then,}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{{x^2} + xy - 2{y^2}}}{{2{x^2} - xy - {y^2}}} = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{\left( {x + 2y} \right)\left( {x - y} \right)}}{{\left( {2x + y} \right)\left( {x - y} \right)}} \cr
& = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{\left( {x + 2y} \right)}}{{\left( {2x + y} \right)}} \cr
& {\text{Evaluating the limit}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{\left( {x + 2y} \right)}}{{\left( {2x + y} \right)}} = \frac{{1 + 2\left( 1 \right)}}{{2\left( 1 \right) + 1}} = \frac{3}{3} = 1 \cr
& {\text{Thus,}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{{x^2} + xy - 2{y^2}}}{{2{x^2} - xy - {y^2}}} = 1 \cr} $$
