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The function [tex]$f$[/tex] is defined over the real numbers. This table gives several values of [tex]$f$[/tex].

\begin{tabular}{rrrrr}
[tex]$x$[/tex] & -2.999 & -2.99 & -2.9 & -2.75 \\
[tex]$f(x)$[/tex] & 3.606 & 3.607 & 3.619 & 3.64
\end{tabular}

Is the table appropriate for approximating [tex]$\lim _{x \rightarrow-3} f(x)$[/tex]? If not, why?

Choose one answer:
A. The table is appropriate.
B. The table isn't appropriate. The [tex]$x$[/tex]-values only approach -3 from one direction.
C. The table isn't appropriate. The increments in [tex]$x$[/tex]-values are constant.



Answer :

GinnyAnswer
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.

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