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Question
Math
Posted 2 months ago

Given the function gg and asked to estimate limx4g(x)\lim _{x \rightarrow 4} g(x), three students created tables.

Each table is accurate, but which one is the best for approximating the limit?

Choose 1 answer:
(A) \begin{tabular}{|crccccc|}
\hline A & xx & 3.9 & 3.95 & 3.999 & 4.001 & 4.05 \\
\hline & g(x)g(x) & -4.475 & -4.488 & -4.5 & -4.5 & -4.513 \\
\hline
\end{tabular}
(B) \begin{tabular}{|cccccccc|}
\hline B & xx & -4.1 & -4.05 & -4.001 & 4.001 & 4.05 & 4.1 \\
\hline & g(x)g(x) & -5.525 & -5.513 & -5.5 & -4.5 & -4.513 & -4.525 \\
\hline
\end{tabular}
(C) \begin{tabular}{|ccccccc|}
\hline C\mathrm{C} & xx & 3 & 3.5 & 3.75 & 3.95 & 3.999 \\
\hline & g(x)g(x) & -4.25 & -4.375 & -4.438 & -4.498 & -4.5 \\
& & & & & &
\end{tabular}
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 2
Table A: x=3.9,3.95,3.999,4.001,4.05x = 3.9, 3.95, 3.999, 4.001, 4.05 and g(x)=4.475,4.488,4.5,4.5,4.513g(x) = -4.475, -4.488, -4.5, -4.5, -4.513
step 3
Table B: x=4.1,4.05,4.001,4.001,4.05,4.1x = -4.1, -4.05, -4.001, 4.001, 4.05, 4.1 and g(x)=5.525,5.513,5.5,4.5,4.513,4.525g(x) = -5.525, -5.513, -5.5, -4.5, -4.513, -4.525
step 4
Table C: x=3,3.5,3.75,3.95,3.999x = 3, 3.5, 3.75, 3.95, 3.999 and g(x)=4.25,4.375,4.438,4.498,4.5g(x) = -4.25, -4.375, -4.438, -4.498, -4.5
step 5
To find the best approximation, we look for the table where g(x)g(x) values are closest to each other as xx approaches 4
step 6
Table A shows g(x)g(x) values converging to 4.5-4.5 as xx approaches 4
step 7
Table B has a wider range of xx values and g(x)g(x) values are not as consistent
step 8
Table C also shows g(x)g(x) values converging to 4.5-4.5, but it has fewer points near x=4x = 4
step 9
Therefore, Table A provides the best approximation for limx4g(x) \lim_{x \rightarrow 4} g(x)
A
Key Concept
Approximating Limits Using Tables
Explanation
The best table for approximating a limit is the one where the function values are most consistent and closest to the limit as the variable approaches the given point.

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