Question

Math

Posted 2 months ago

```
Given the function $g$ and asked to estimate $\lim _{x \rightarrow 4} g(x)$, 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 & $x$ & 3.9 & 3.95 & 3.999 & 4.001 & 4.05 \\
\hline & $g(x)$ & -4.475 & -4.488 & -4.5 & -4.5 & -4.513 \\
\hline
\end{tabular}
(B) \begin{tabular}{|cccccccc|}
\hline B & $x$ & -4.1 & -4.05 & -4.001 & 4.001 & 4.05 & 4.1 \\
\hline & $g(x)$ & -5.525 & -5.513 & -5.5 & -4.5 & -4.513 & -4.525 \\
\hline
\end{tabular}
(C) \begin{tabular}{|ccccccc|}
\hline $\mathrm{C}$ & $x$ & 3 & 3.5 & 3.75 & 3.95 & 3.999 \\
\hline & $g(x)$ & -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: $x = 3.9, 3.95, 3.999, 4.001, 4.05$ and $g(x) = -4.475, -4.488, -4.5, -4.5, -4.513$

step 3

Table B: $x = -4.1, -4.05, -4.001, 4.001, 4.05, 4.1$ and $g(x) = -5.525, -5.513, -5.5, -4.5, -4.513, -4.525$

step 4

Table C: $x = 3, 3.5, 3.75, 3.95, 3.999$ and $g(x) = -4.25, -4.375, -4.438, -4.498, -4.5$

step 5

To find the best approximation, we look for the table where $g(x)$ values are closest to each other as $x$ approaches 4

step 6

Table A shows $g(x)$ values converging to $-4.5$ as $x$ approaches 4

step 7

Table B has a wider range of $x$ values and $g(x)$ values are not as consistent

step 8

Table C also shows $g(x)$ values converging to $-4.5$, but it has fewer points near $x = 4$

step 9

Therefore, Table A provides the best approximation for $\lim_{x \rightarrow 4} g(x)$

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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