Question

the following tables, which best reflects the values of a function g for which (lim_{x \to 7} g(x) = 6)? tables a, b, c, d with x and g(x) values as shown: table a: x=5.85,5.90,5.95,5.99,6.01,6.05,6.10,6.15; g(x)=7.126,7.075,7.033,7.006,6.995,6.977,6.964,6.960. table b: x=6.85,6.90,6.95,6.99,7.01,7.05,7.10,7.15; g(x)=5.620,5.837,5.961,5.998,5.990,5.964,5.863,5.709. table c: x=6.85,6.90,6.95,6.99,7.01,7.05,7.10,7.15; g(x)=5.919,5.942,5.969,5.993,7.017,7.087,7.177,7.269. table d: x=6.85,6.90,6.95,6.99,7.01,7.05,7.10,7.15; g(x)=1.362,5.954,10.691,14.690,6.010,6.049,6.095,6.149

Explanation:

Step1: Recall the limit definition

For $\lim_{x \to a} g(x) = L$, as $x$ approaches $a$ from both left and right, $g(x)$ should approach $L$. Here $a = 7$, $L = 6$. So we check $x$ values near 7 (left: $x < 7$, right: $x > 7$) and see if $g(x)$ approaches 6.

Step2: Analyze Option A

In Option A, $x$ values are around 6 (5.85, 5.90, ..., 6.15), not near 7. So it's about $\lim_{x \to 6}$, not $\lim_{x \to 7}$. Eliminate A.

Step3: Analyze Option B

For $x$ approaching 7 from left (6.85, 6.90, 6.95, 6.99): $g(x)$ values are 5.620, 5.837, 5.961, 5.998 (approaching 6). From right (7.01, 7.05, 7.10, 7.15): $g(x)$ values are 5.990, 5.964, 5.863, 5.709 (approaching 6, since left and right both approach 6).

Step4: Analyze Option C

From left of 7: $g(x)$ approaches ~6 (5.919, 5.942, 5.969, 5.993). From right: $g(x)$ is 7.017, 7.087, ... (approaching 7, not 6). So left and right limits don't match 6. Eliminate C.

Step5: Analyze Option D

$g(x)$ values are inconsistent (1.362, 10.691, 14.690, ...) and don't approach 6. Eliminate D.

Thus, Option B best reflects $\lim_{x \to 7} g(x) = 6$.

Answer:

B