Question

the table above gives values of the function f at selected values of x. which of the following statements must be true?
a) $lim_{x \to 1} f(x) = 2$
b) $lim_{x \to 1} f(x) = 7$
c) $lim_{x \to 1} f(x)$ does not exist.
d) $lim_{x \to 1} f(x)$ cannot be definitively determined from the data in the table.
(table: x values: 0.9, 0.99, 0.999, 0.9999, 1, 1.0001, 1.001, 1.01, 1.1; f(x) values: 6.80, 6.86, 6.90, 6.95, 2, 7.05, 7.10, 7.14, 7.20)

Explanation:

To determine the limit as $x$ approaches 1, we analyze the left - hand limit (as $x$ approaches 1 from values less than 1) and the right - hand limit (as $x$ approaches 1 from values greater than 1).

Left - hand limit analysis:

As $x$ approaches 1 from the left (values like 0.9, 0.99, 0.999, 0.9999), the values of $f(x)$ are 6.80, 6.86, 6.90, 6.95. These values seem to be approaching 7 (since 6.80, 6.86, 6.90, 6.95 are getting closer to 7 as $x$ gets closer to 1 from the left).

Right - hand limit analysis:

As $x$ approaches 1 from the right (values like 1.0001, 1.001, 1.01, 1.1), the values of $f(x)$ are 7.05, 7.10, 7.14, 7.20. These values also seem to be approaching 7 as $x$ gets closer to 1 from the right. But the value of the function at $x = 1$ is $f(1)=2$. However, the limit of a function as $x$ approaches a point does not depend on the value of the function at that point. But the problem is that we only have a finite set of sample points approaching 1 from the left and right. We don't know the behavior of the function in the entire neighborhood around $x = 1$ (for example, what happens for values of $x$ very close to 1, say $x=0.99999$ or $x = 1.00001$ that are not in the table). So we cannot be completely certain that the left - hand limit and the right - hand limit are exactly 7 or that they are equal. It is possible that the function could behave in an unexpected way between the given sample points. So we cannot definitively determine the limit from the given table data.

Option A is incorrect because the value of the function at $x = 1$ is not relevant to the limit, and the left - hand and right - hand approaching values are not approaching 2. Option B is incorrect because even though the sample points seem to approach 7, we don't have enough information (we don't know the function's behavior for all $x$ near 1) to be certain that the limit is 7. Option C is incorrect because from the given sample points, the left - hand and right - hand approaches seem to be converging to the same general value (around 7), so we can't say the limit does not exist based on this data.

Answer:

D. $\lim\limits_{x \to 1} f(x)$ cannot be definitively determined from the data in the table.