Answer
The limit $\underset{x\to 2}{\mathop{\lim }}\,\sqrt{5x-6}$ is 2. And the corresponding limit property is $\underset{x\to a}{\mathop{\lim }}\,\sqrt[n]{f\left( x \right)}=\sqrt[n]{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}=\sqrt[n]{L}$, where $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L $ and $ n $ is any integer greater than or equal to 2 provided that all roots represent real numbers.
Work Step by Step
Consider the given limit $\underset{x\to 2}{\mathop{\lim }}\,\sqrt{5x-6}$.
First, find $\underset{x\to 2}{\mathop{\lim }}\,\left( 5x-6 \right)$.
Use the limit properties $\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)-g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)-\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$ and $\underset{x\to a}{\mathop{\lim }}\,c=c $, where $ c $ is a constant.
$\begin{align}
& \underset{x\to 2}{\mathop{\lim }}\,\left( 5x-6 \right)=\underset{x\to 2}{\mathop{\lim }}\,5x-\underset{x\to 2}{\mathop{\lim }}\,6 \\
& =10-6 \\
& =4
\end{align}$
The limit that is required is calculated by taking this limit 4, and taking its square root.
Thus, $\underset{x\to 2}{\mathop{\lim }}\,\sqrt{\left( 5x-6 \right)}=\sqrt{4}=2$
In limit notation, the corresponding limit property is
If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L $ and $ n $ is any integer greater than or equal to 2, then
$\underset{x\to a}{\mathop{\lim }}\,\sqrt[n]{f\left( x \right)}=\sqrt[n]{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}=\sqrt[n]{L}$, provided that all roots represent real numbers.
In words, the limit of the nth root of a function is found by taking the limit of the function and then taking the nth root of this limit.
