Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 11

Show that the function $f(x) = (x + 2x^3)^4$ is continuous at the number $a = -1$ using the definition of continuity and the properties of limits.

By using properties of limit,

$
\begin{equation}
\begin{aligned}
& \lim \limits_{x \to -1} (x + 2x^3)^4 && = (\lim \limits_{x \to -1} x + 2 \lim \limits_{x \to -1} x^3)^4
&& \text{Apply sum and power law}\\
\\
& \phantom{x} && = [-1 + 2(-1)^3]^4
&& \text{Substitute the given value of $a$}\\
& \phantom{x} &&= 81

\end{aligned}
\end{equation}
$


By using the definition of continuity,
$\lim \limits_{x \to a} f(x) = f(a)$




$
\begin{equation}
\begin{aligned}
& \lim \limits_{x \to -1} (x+2x^3)^4 && = f(-1) = [-1+2(-1)^3]^4\\
& \phantom{x} && = 81
\end{aligned}
\end{equation}
$


Therefore, by applying either of the two, we have shown that the function is continuous at -1 and is equal to 81