# Chapter 11 - Section 11.2 - Finding Limits Using Properties of Limits - Concept and Vocabulary Check - Page 1153: 8

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”.

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