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 \to 3} f(x)\\)? \
\
a) 0 \
b) 3 \
c) 5 \
d) there is no best approximation, because the limit does not exist \
\
(table: x values: 2.9, 2.99, 2.999, 3.001, 3.01, 3.1; f(x) values: 5.018, 5.007, 5.002, 4.998, 4.982, 4.887)

Explanation:

Step1: Analyze left - hand limit

As \(x\) approaches \(3\) from the left (values like \(2.9\), \(2.99\), \(2.999\)), we look at the corresponding \(f(x)\) values: \(f(2.9) = 5.018\), \(f(2.99)=5.007\), \(f(2.999) = 5.002\). As \(x\) gets closer to \(3\) from the left, \(f(x)\) gets closer to \(5\).

Step2: Analyze right - hand limit

As \(x\) approaches \(3\) from the right (values like \(3.001\), \(3.01\), \(3.1\)), we look at the corresponding \(f(x)\) values: \(f(3.001)=4.998\), \(f(3.01) = 4.982\), \(f(3.1)=4.887\). As \(x\) gets closer to \(3\) from the right, \(f(x)\) gets closer to \(5\).

Step3: Determine the limit

Since the left - hand limit (as \(x\to3^{-}\)) and the right - hand limit (as \(x\to3^{+}\)) both approach \(5\), by the definition of the limit of a function, \(\lim_{x\to3}f(x)=5\).

Answer:

C. 5