To determine whether the table is appropriate for approximating [tex]\(\lim_{x \rightarrow -3} f(x)\)[/tex], we need to consider how the values of [tex]\(x\)[/tex] are approaching [tex]\(-3\)[/tex].
According to the table:
[tex]\[
\begin{array}{cccc}
x & -2.999 & -2.99 & -2.9 & -2.75 \\
f(x) & 3.606 & 3.607 & 3.619 & 3.64 \\
\end{array}
\][/tex]
Let's analyze the [tex]\(x\)[/tex]-values more closely:
- -2.999
- -2.99
- -2.9
- -2.75
All of these values are greater than [tex]\(-3\)[/tex] and are approaching [tex]\(-3\)[/tex] from the right (i.e., from values greater than [tex]\(-3\)[/tex]).
To properly approximate [tex]\(\lim_{x \rightarrow -3} f(x)\)[/tex], the [tex]\(x\)[/tex]-values should approach [tex]\(-3\)[/tex] from both directions: values slightly less than [tex]\(-3\)[/tex] and values slightly greater than [tex]\(-3\)[/tex]. However, in this table, the [tex]\(x\)[/tex]-values only approach [tex]\(-3\)[/tex] from the right.
Thus, because the [tex]\(x\)[/tex]-values are not approaching [tex]\(-3\)[/tex] from both sides, the table is not appropriate for approximating the limit as [tex]\(x\)[/tex] approaches [tex]\(-3\)[/tex]. Therefore, the correct reason why the table is not appropriate for approximating [tex]\(\lim_{x \rightarrow -3} f(x)\)[/tex] is:
(B) The table isn't appropriate. The [tex]$x$[/tex]-values only approach -3 from one direction.
