Question

the table above gives selected values for a continuous function f. based on the data in the table, which of the following is the best approximation for lim_{x→4}f(x)? a 0 b 4 c 7 d there is no best approximation, because the limit does not exist.

Explanation:

Step1: Recall limit definition for continuous functions

For a continuous function $f(x)$, $\lim_{x
ightarrow a}f(x)=f(a)$. We can approximate the limit as $x$ approaches a value by looking at values of $f(x)$ as $x$ gets closer to that value from both the left - hand side and the right - hand side.

Step2: Analyze left - hand limit

As $x$ approaches $4$ from the left - hand side ($x = 3.9,3.99,3.999$), the values of $f(x)$ are $7.018,7.007,7.002$ respectively. These values are approaching $7$.

Step3: Analyze right - hand limit

As $x$ approaches $4$ from the right - hand side ($x = 4.001,4.01,4.1$), the values of $f(x)$ are $6.998,5.982,5.887$. The value of $f(x)$ for $x = 4.001$ is very close to $7$. Since the function is continuous, we consider the values closest to $x = 4$ on both sides. The values of $f(x)$ for $x$ values very close to $4$ (like $3.999$ and $4.001$) are approximately $7$.

Answer:

C. 7