Answer
The complete statement is “$\underset{x\to a}{\mathop{\lim }}\,\text{ }\sqrt[n]{f\left( x \right)}=$$\sqrt[n]{L}$ where $ n\ge 2$ is an integer and all roots represent real numbers”.
Work Step by Step
In case of the limit of a root, find the limit of the function and then take the $ n\text{th}$ root of the limit. That is:
The limit of a root:
If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L $ and $ n $ is a positive integer greater than or equal to $2$, then
$\underset{x\to a}{\mathop{\lim }}\,\sqrt[n]{f\left( x \right)}=\sqrt[n]{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}=\sqrt[n]{L}$
Where all roots represent real numbers.
The limit of the $ n\text{th}$ root of a function is found by taking the limit of the function and then taking the $ n\text{th}$ root of this limit.
For example: Let $ f\left( x \right)=x $,
$\begin{align}
& \underset{x\to 7}{\mathop{\lim }}\,\sqrt[3]{f\left( x \right)}=\sqrt[3]{\underset{x\to 7}{\mathop{\lim }}\,f\left( x \right)} \\
& =\sqrt[3]{\underset{x\to 7}{\mathop{\lim }}\,x} \\
& =\sqrt[3]{7}
\end{align}$
Therefore, the complete fill for the blank in the statement “$\underset{x\to a}{\mathop{\lim }}\,\text{ }\sqrt[n]{f\left( x \right)}=\sqrt[n]{L}$ where $ n\ge 2$ is an integer and all roots represent real numbers”.
