Answer
$2$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{u \to - \infty } \frac{{\left( {{u^2} + 1} \right)\left( {2{u^2} - 1} \right)}}{{{{\left( {{u^2} + 2} \right)}^2}}} \cr
& {\text{Simplify the numerator and expand the denominator}} \cr
& = \mathop {\lim }\limits_{u \to - \infty } \frac{{2{u^4} - {u^2} + 2{u^2} - 1}}{{{u^4} + 4{u^2} + 4}} \cr
& = \mathop {\lim }\limits_{u \to - \infty } \frac{{2{u^4} + {u^2} - 1}}{{{u^4} + 4{u^2} + 4}} \cr
& {\text{Use properties of limits}} \cr
& = \frac{{\mathop {\lim }\limits_{u \to - \infty } \left( {2{u^4} + {u^2} - 1} \right)}}{{\mathop {\lim }\limits_{u \to - \infty } \left( {{u^4} + 4{u^2} + 4} \right)}} \cr
& = \frac{{\mathop {\lim }\limits_{u \to - \infty } \left( {2{u^4}} \right) + \mathop {\lim }\limits_{u \to - \infty } \left( {{u^2}} \right) - \mathop {\lim }\limits_{u \to - \infty } \left( 1 \right)}}{{\mathop {\lim }\limits_{u \to - \infty } \left( {{u^4}} \right) + \mathop {\lim }\limits_{u \to - \infty } \left( {4{u^2}} \right) + \mathop {\lim }\limits_{u \to - \infty } \left( 4 \right)}} = \frac{\infty }{\infty }{\text{ Ind}} \cr
& {\text{Divide the numerator and denominator by }}{u^4} \cr
& \mathop {\lim }\limits_{u \to - \infty } \frac{{2{u^4} + {u^2} - 1}}{{{u^4} + 4{u^2} + 4}} = \mathop {\lim }\limits_{u \to - \infty } \frac{{\frac{{2{u^4}}}{{{u^4}}} + \frac{{{u^2}}}{{{u^4}}} - \frac{1}{{{u^4}}}}}{{\frac{{{u^4}}}{{{u^4}}} + \frac{{4{u^2}}}{{{u^4}}} + \frac{4}{{{u^4}}}}} \cr
& = \mathop {\lim }\limits_{u \to - \infty } \frac{{2 + \frac{1}{{{u^2}}} - \frac{1}{{{u^4}}}}}{{1 + \frac{4}{{{u^2}}} + \frac{4}{{{u^4}}}}} \cr
& {\text{Use properties of limits}} \cr
& = \frac{{\mathop {\lim }\limits_{u \to - \infty } \left( 2 \right) + \mathop {\lim }\limits_{u \to - \infty } \left( {\frac{1}{{{u^2}}}} \right) - \mathop {\lim }\limits_{u \to - \infty } \left( {\frac{1}{{{u^4}}}} \right)}}{{\mathop {\lim }\limits_{u \to - \infty } \left( 1 \right) + \mathop {\lim }\limits_{u \to - \infty } \left( {\frac{4}{{{u^2}}}} \right) + \mathop {\lim }\limits_{u \to - \infty } \left( {\frac{4}{{{u^4}}}} \right)}} \cr
& {\text{Evaluate the limits}} \cr
& = \frac{{2 + 0 - 0}}{{1 + 0 + 0}} \cr
& = 2 \cr
& {\text{Therefore}}{\text{,}} \cr
& \mathop {\lim }\limits_{u \to - \infty } \frac{{\left( {{u^2} + 1} \right)\left( {2{u^2} - 1} \right)}}{{{{\left( {{u^2} + 2} \right)}^2}}} = 2 \cr} $$
