To estimate the limit [tex]\(\lim_{x \rightarrow -8^{+}} g(x)\)[/tex], we need to carefully consider the values of [tex]\(g(x)\)[/tex] as [tex]\(x\)[/tex] approaches [tex]\(-8\)[/tex] from the right (i.e., values of [tex]\(x\)[/tex] that are slightly greater than [tex]\(-8\)[/tex]).
Given the provided table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & g(x) \\
\hline
-8.025 & -5.2 \\
-8.005 & -5.04 \\
-8.001 & -5.008 \\
-8 & 8 \\
-7.999 & 0.008 \\
-7.995 & 0.04 \\
-7.975 & 0.2 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Identify the values of [tex]\(x\)[/tex] that are just greater than [tex]\(-8\)[/tex]:
- We need [tex]\(x\)[/tex] values that are slightly more than [tex]\(-8\)[/tex]. In the table, those values are [tex]\(-7.999\)[/tex], [tex]\(-7.995\)[/tex], and [tex]\(-7.975\)[/tex].
2. Extract the corresponding [tex]\(g(x)\)[/tex] values:
- For [tex]\(x = -7.999\)[/tex], [tex]\(g(x) = 0.008\)[/tex]
- For [tex]\(x = -7.995\)[/tex], [tex]\(g(x) = 0.04\)[/tex]
- For [tex]\(x = -7.975\)[/tex], [tex]\(g(x) = 0.2\)[/tex]
3. Average these [tex]\(g(x)\)[/tex] values to estimate the limit:
- Calculate the average of [tex]\(g(x)\)[/tex] values: [tex]\[ g(-7.999), g(-7.995), g(-7.975) \][/tex]
[tex]\[
\text{Average} = \frac{0.008 + 0.04 + 0.2}{3}
\][/tex]
4. Perform the calculation:
- Summing the values: [tex]\(0.008 + 0.04 + 0.2 = 0.248\)[/tex]
- Dividing by the number of values: [tex]\(\frac{0.248}{3} \approx 0.08267\)[/tex]
### Final Estimate:
[tex]\[
\lim_{x \rightarrow -8^{+}} g(x) \approx 0.08267
\][/tex]
Thus, a reasonable estimate for the limit [tex]\(\lim_{x \rightarrow -8^{+}} g(x)\)[/tex] based on the given data is approximately [tex]\(0.08267\)[/tex].
