Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 8

Determine the $\lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6}$ and justify each step by indicating the appropriate limit law(s).


$\begin{equation} \begin{aligned} \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2} (u^4+3u+6)} && \text{(Root Law)}\\ \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2} u^4 + \lim\limits_{u \rightarrow -2}3u + \lim\limits_{u \rightarrow -2} 6} && \text{(Sum Law)}\\ \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{\lim\limits_{u \rightarrow -2}u^4 + 3 \lim\limits_{u \rightarrow -2} u + 6 } && \text{(Constant Multiple and Constant Law)}\\ \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} & = \sqrt{(-2)^4+3(-2)+6} && \text{(Power Special Limit Law)} \end{aligned} \end{equation}\\ \boxed{ \lim\limits_{u \rightarrow -2} \quad \sqrt{u^4+3u+6} = 4}$