Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 6

Determine the
Given: $\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5$ and justify each step by indicating the appropriate limit law(s).


$
\begin{equation}
\begin{aligned}
\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 &= \lim\limits_{t \rightarrow -1} (t^2+1)^3 \cdot
\lim\limits_{t \rightarrow -1} (t+3)^5
&& \text{(Product Law)}\\
\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 &= \left[
\lim\limits_{t \rightarrow -1} (t^2+1)
\right]^3
\left[
\lim\limits_{t \rightarrow -1} (t+3)
\right]^5
&& \text{(Power Law)}\\
\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 &= \left(
\lim\limits_{t \rightarrow -1} t^2 +
\lim\limits_{t \rightarrow -1} 1
\right)^3
\left(
\lim\limits_{t \rightarrow -1} t +
\lim\limits_{t \rightarrow -1} 3
\right)^5
&& \text{(Sum Law)}\\
\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 &= \left(
\lim\limits_{t \rightarrow -1} t^2+1
\right)^3
\left(
\lim\limits_{t \rightarrow -1} t+3
\right)^5
&& \text{(Constant Law)}\\
\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 &= [(-1)^2+1]^3[(-1)+3]^5
&& \text{(Power Special Limit Law)}
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{t \rightarrow -1} \quad (t^2+1)^3(t+3)^5 = 256}
$