Question

which of the statements a through i about the function ( y = f(x) ) graphed here are true, and which are false? a. the statement ( lim_{x \to 2} f(x) ) does not exist is false. b. the statement ( lim_{x \to 2} f(x) = 2 ) is

Explanation:

Step1: Recall the definition of the limit

The limit of a function \(y = f(x)\) as \(x\to a\) is the value that \(f(x)\) approaches as \(x\) gets arbitrarily close to \(a\) (not necessarily equal to \(a\)). Mathematically, \(\lim_{x\to a}f(x)=L\) if \(\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x) = L\)

Step2: Analyze the left - hand and right - hand limits as \(x\to2\)

As \(x\) approaches \(2\) from the left (\(x\to2^{-}\)) and from the right (\(x\to2^{+}\)), we observe the \(y\) - values of the function. Looking at the graph, as \(x\) approaches \(2\) (both from the left and the right), the \(y\) - value of the function approaches \(1\).

Answer:

The statement \(\lim_{x\to2}f(x) = 2\) is false.