Answer
The correct answers to fill the blanks provided in the statement “We find $\underset{x\to {{2}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ using $ f\left( x \right)=$_______ and we find $\underset{x\to {{2}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ using $ f\left( x \right)=$_______” are ${{x}^{2}}+5$ and ${{x}^{3}}+1$ respectively.
Work Step by Step
Consider the piecewise function defined by:
$ f\left( x \right)=\left\{ \begin{align}
& {{x}^{2}}+5\text{ if }x<2 \\
& {{x}^{3}}+1\text{ if }x\ge 2 \\
\end{align} \right.$
To find $\underset{x\to {{2}^{-}}}{\mathop{\lim }}\,f\left( x \right)$, look at values of $ f\left( x \right)$ when $ x $ is close to $2$ but less than $2$. Because $ x $ is less than $2$, use the first line of the piecewise function’s equation, $ f\left( x \right)={{x}^{2}}+5$ where $ x<2$.
Similarly, to find $\underset{x\to {{2}^{+}}}{\mathop{\lim }}\,f\left( x \right)$, look at values of $ f\left( x \right)$ when $ x $ is close to $2$ but greater than $2$. Because $ x $ is greater than $2$, use the second line of the piecewise function’s equation, $ f\left( x \right)={{x}^{3}}+1$ where $ x\ge 2$.
Therefore, the complete statement with filled blanks would be “We find $\underset{x\to {{2}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ using $ f\left( x \right)=$${{x}^{2}}+5$ and we find $\underset{x\to {{2}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ using $ f\left( x \right)=$${{x}^{3}}+1$”.
