Answer
The complete statement is “$\underset{x\to a}{\mathop{\lim }}\,\text{ }{{\left[ f\left( x \right) \right]}^{n}}=$${{L}^{n}}$ where $ n\ge 2$ is an integer”.
Work Step by Step
In case of the limit of a power, find the limit of the function and then raise the limit to the power. That is:
The limit of a power:
If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L $ and $ n $ is a positive integer, then
$\underset{x\to a}{\mathop{\lim }}\,{{\left[ f\left( x \right) \right]}^{n}}={{\left[ \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right]}^{n}}={{L}^{n}}$
The limit of the function $ f\left( x \right)$ as $ x $ approaches $ a $ is the function evaluated at $ a $. Because this function is raised to the $ n\text{th}$ power, the limit that we seek is the limit of the function raised to the $ n\text{th}$ power.
For example: Let $ f\left( x \right)=x $,
$\begin{align}
& \underset{x\to 7}{\mathop{\lim }}\,{{\left[ f\left( x \right) \right]}^{2}}={{\left[ \underset{x\to 7}{\mathop{\lim }}\,f\left( x \right) \right]}^{2}} \\
& ={{\left[ \underset{x\to 7}{\mathop{\lim }}\,x \right]}^{2}} \\
& ={{\left[ 7 \right]}^{2}} \\
& =49
\end{align}$
Therefore, the complete fill for the blank in the statement “ $\underset{x\to a}{\mathop{\lim }}\,\text{ }{{\left[ f\left( x \right) \right]}^{n}}=$${{L}^{n}}$, where $ n\ge 2$ is an integer”.
