Question

use the given graph of f to state the value of each quantity, if it exists. (if an answer does not exist, enter dne.)
(a) \\(\lim_{x \to 2^-} f(x)\\)
(b) \\(\lim_{x \to 2^+} f(x)\\)
(c) \\(\lim_{x \to 2} f(x)\\)
(d) \\(f(2)\\)
(e) \\(\lim_{x \to 4} f(x)\\)
(f) \\(f(4)\\)

Explanation:

Part (a)

Step1: Analyze left - hand limit as \(x\to2^{-}\)

To find \(\lim_{x\to2^{-}}f(x)\), we look at the values of \(f(x)\) as \(x\) approaches 2 from the left - hand side (values of \(x\) less than 2). From the graph, as \(x\) gets closer to 2 from the left, the \(y\) - value (the value of \(f(x)\)) approaches 3.

Step1: Analyze right - hand limit as \(x\to2^{+}\)

To find \(\lim_{x\to2^{+}}f(x)\), we look at the values of \(f(x)\) as \(x\) approaches 2 from the right - hand side (values of \(x\) greater than 2). From the graph, as \(x\) gets closer to 2 from the right, the \(y\) - value (the value of \(f(x)\)) approaches 1.

Step1: Recall the condition for the existence of a limit

For \(\lim_{x\to a}f(x)\) to exist, \(\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)\). We found that \(\lim_{x\to2^{-}}f(x) = 3\) and \(\lim_{x\to2^{+}}f(x)=1\). Since \(3
eq1\), the two - sided limit \(\lim_{x\to2}f(x)\) does not exist.

Answer:

\(3\)

Part (b)