Answer
The complete statement is “$\underset{x\to a}{\mathop{\lim }}\,\text{ }\left[ f\left( x \right)\cdot g\left( x \right) \right]=$$ LM $.”
Work Step by Step
In case of the limit of a product, find the limit of each function in the product and then take the product of each of the limits. That is,
The limit of a product:
If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=L\text{ and }\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=M $, then
$\underset{x\to a}{\mathop{\lim }}\,\left[ f\left( x \right)\cdot g\left( x \right) \right]\underset{x\to a}{\mathop{=\lim }}\,f\left( x \right)\cdot \underset{x\to a}{\mathop{\lim }}\,g\left( x \right)=LM $.
The limit of the product of two functions equals the product of their limits.
For example:
Let $ f\left( x \right)=x $ and $ g\left( x \right)=2$,
$\begin{align}
& \underset{x\to 7}{\mathop{\lim }}\,\left[ f\left( x \right)\cdot g\left( x \right) \right]=\underset{x\to 7}{\mathop{\lim }}\,f\left( x \right)\cdot \underset{x\to 7}{\mathop{\lim }}\,g\left( x \right) \\
& =\underset{x\to 7}{\mathop{\lim }}\,x\cdot \underset{x\to 7}{\mathop{\lim }}\,2 \\
& =7\cdot 2 \\
& =14
\end{align}$
Therefore, the complete fill for the blank in the statement “$\underset{x\to a}{\mathop{\lim }}\,\text{ }\left[ f\left( x \right)\cdot g\left( x \right) \right]=$$ LM $.”
