Use the definition of continuity and the properties of limits to show that the function...

Limits
Limit of the function at point a exists if and only if left hand limit and right hand limit are equal and exists finitely at point a.
Left hand limit
${\displaystyle \underset{x\to a^{-}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a-h)}$
Right hand limit
${\displaystyle \underset{x\to a^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a+h)}$
So, limit exists at pont a if and only if ${\displaystyle \underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=}$ finite
Continuity
A function is said to be continuous at a if and only if
1. Limit exits that is ${\displaystyle \underset{x\to a}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=}$ finite
2. Limit is equal to value of the function at point a that is ${\displaystyle \underset{x\to a^{-}}{lim}f\left(x\right)=\underset{x\to a^{+}}{lim}f\left(x\right)=f\left(a\right)}$
Solution:
We have given function ${\displaystyle f\left(x\right)={(x+2x^3)}^4}$ at points a=-1
Left hand limit
${\displaystyle \underset{x\to a^{+}}{lim}f\left(x\right)=\underset{x\to -1^{+}}{lim}{(x+2x^3)}^4=\underset{h\to 0}{lim}{((-1+h)+2{(-1+h)}^3)}^4}$
${\displaystyle ={(-1+2{(-1)}^3)}^4}$
${\displaystyle ={(-1-2)}^4={(-3)}^4=81}$
Hence ${\displaystyle \underset{x\to -1^{+}}{lim}f\left(x\right)=81}$
So limit exists
Now, ${\displaystyle f(-1)={(-1+2{(-1)}^3)}^4}$
${\displaystyle ={(-1-2)}^4={(-3)}^4=81}$
Therefore,
Given function is continuous at x=-1