Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 5

Determine the $\lim\limits_{x \rightarrow 8} \quad (1+\sqrt[3]{x})(2-6x^2+x^3)$ and justify each step by indicating the appropriate limit law(s).



$
\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow 8} \quad (1+\sqrt[3]{x})(2-6x^2+x^3) &= \lim\limits_{x \rightarrow 8} (1+x^{\frac{1}{3}}) \cdot
\lim\limits_{x \rightarrow 8} (2-6x^2+x^3)
&& \text{(Product Law)}\\
\lim\limits_{x \rightarrow 8} \quad (1+\sqrt[3]{x})(2-6x^2+x^3) &= \lim\limits_{x \rightarrow 8} \left(
\lim\limits_{x \rightarrow 8} 1 +
\lim\limits_{x \rightarrow 8} x^{\frac{1}{3}}
\right)
\left(
\lim\limits_{x \rightarrow 8} 2 -
6\lim\limits_{x \rightarrow 8} x^2 +
\lim\limits_{x \rightarrow 8} x^3
\right)
&& \text{(Sum, Difference and Constant Multiple Law)}\\
\lim\limits_{x \rightarrow 8} \quad (1+\sqrt[3]{x})(2-6x^2+x^3) &= \left[
1+(8)^{\frac{1}{3}}
\right]
\left[
2-6(8)^2+(8)^3
\right]
&& \text{(Constant, Power Special Limit Law)}
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \rightarrow 8} \quad (1+\sqrt[3]{x})(2-6x^2+x^3) = 390}
$