Answer
$5$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{u \to - 2} \sqrt {9 - {u^3} + 2{u^2}} \cr
& {\text{Suppose that }}c{\text{ is a constant and the limits }} \cr
& {\text{ }}\mathop {\lim }\limits_{x \to a} f\left( x \right){\text{ and }}\mathop {\lim }\limits_{x \to a} g\left( x \right) \cr
& \cr
& {\text{Use the Root Law }}\mathop {\lim }\limits_{x \to a} \root n \of {f\left( x \right)} = \root n \of {\mathop {\lim }\limits_{x \to a} f\left( x \right)} ,{\text{ }}n{\text{ }} \in {\text{ Z + }} \cr
& \mathop {\lim }\limits_{u \to - 2} \sqrt {9 - {u^3} + 2{u^2}} = \sqrt {\mathop {\lim }\limits_{u \to - 2} \left( {9 - {u^3} + 2{u^2}} \right)} \cr
& \cr
& {\text{Use the Sum and Difference Laws }} \cr
& \mathop {\lim }\limits_{x \to a} \left[ {f\left( x \right) \pm g\left( x \right)} \right] = \mathop {\lim }\limits_{x \to a} f\left( x \right) \pm \mathop {\lim }\limits_{x \to a} g\left( x \right) \cr
& \sqrt {\mathop {\lim }\limits_{u \to - 2} \left( {9 - {u^3} + 2{u^2}} \right)} = \sqrt {\mathop {\lim }\limits_{u \to - 2} \left( 9 \right) - \mathop {\lim }\limits_{u \to - 2} \left( {{u^3}} \right) + \mathop {\lim }\limits_{u \to - 2} \left( {2{u^2}} \right)} \cr
& \cr
& {\text{Use the Constant Multiple Law }}\left( {{\text{see page 95}}} \right) \cr
& \mathop {\lim }\limits_{x \to a} \left[ {cf\left( x \right)} \right] = c\mathop {\lim }\limits_{x \to a} f\left( x \right) \cr
& = \sqrt {\mathop {\lim }\limits_{u \to - 2} \left( 9 \right) - \mathop {\lim }\limits_{u \to - 2} \left( {{u^3}} \right) + 2\mathop {\lim }\limits_{u \to - 2} \left( {{u^2}} \right)} \cr
& \cr
& {\text{Use The power Law }}\mathop {\lim }\limits_{x \to a} {x^n} = {a^n}{\text{, }}n{\text{ is a positive integer }} \cr
& {\text{and the laws }}\mathop {\lim }\limits_{x \to a} x = a,{\text{ }}\mathop {\lim }\limits_{x \to a} c = c,{\text{ then}} \cr
& = \sqrt {9 - {{\left( { - 2} \right)}^3} + 2{{\left( { - 2} \right)}^2}} \cr
& {\text{Simplify}} \cr
& = \sqrt {9 + 8 + 8} \cr
& = 5 \cr} $$
