Answer
The limit $\underset{x\to 2}{\mathop{\lim }}\,{{\left( 3{{x}^{2}}-10 \right)}^{3}}$ is 8. And the corresponding limit property is $\underset{x\to a}{\mathop{\lim }}\,{{\left[ f\left( x \right) \right]}^{n}}={{\left[ \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right]}^{n}}={{L}^{n}}$, where $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L $ and $ n $ is any integer.
Work Step by Step
Consider the given limit, $\underset{x\to 2}{\mathop{\lim }}\,{{\left( 3{{x}^{2}}-10 \right)}^{3}}$.
First, find $\underset{x\to 2}{\mathop{\lim }}\,\left( 3{{x}^{2}}-10 \right)$.
$\begin{align}
& \underset{x\to 2}{\mathop{\lim }}\,\left( 3{{x}^{2}}-10 \right)=\left( 3\times {{2}^{2}} \right)-10 \\
& =3\times 4-10 \\
& =12-10 \\
& =2
\end{align}$
The limit that is required is calculated by taking this limit, 2, and raising it to the third power.
Thus, $\underset{x\to 2}{\mathop{\lim }}\,{{\left( 3{{x}^{2}}-10 \right)}^{3}}={{\left[ \underset{x\to 2}{\mathop{\lim }}\,\left( 3{{x}^{2}}-10 \right) \right]}^{3}}={{2}^{3}}=8$.
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, then
$\underset{x\to a}{\mathop{\lim }}\,{{\left[ f\left( x \right) \right]}^{n}}={{\left[ \underset{x\to a}{\mathop{\lim }}\,f\left( x \right) \right]}^{n}}={{L}^{n}}$
In words, the limit of a function to a power is found by taking the limit of the function and then raising this limit to the power.
