Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 3

Determine the $\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1)$ and justify each step by indicating the appropriate limit law(s).


$
\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1) &= \lim\limits_{x \rightarrow -2} 3x^4 +
\lim\limits_{x \rightarrow -2} 2x^2 -
\lim\limits_{x \rightarrow -2} x +
\lim\limits_{x \rightarrow -2} 1
&& \text{(Sum and difference Law)}\\
\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1) &= 3\lim\limits_{x \rightarrow -2} x^4 +
2\lim\limits_{x \rightarrow -2} x^2 -
\lim\limits_{x \rightarrow -2} x +
\lim\limits_{x \rightarrow -2} 1
&& \text{(Constant Multiple Law)}\\
\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1) &= 3\lim\limits_{x \rightarrow -2} x^4 +
2\lim\limits_{x \rightarrow -2} x^2 -
\lim\limits_{x \rightarrow -2} x +
1
&& \text{(Special Limit, Constant Multiple Law.)}\\
\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1) &= 3(-2)^4 +
2(-2)^2 -
(-2) +
1
&& \text{(Power Special Limit)}
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \rightarrow -2} (3x^4+2x^2-x+1) = 59}
$