Questions: The graph below is the function (f(x)) Determine which one of the following rules for continuity is violated first at (x=2). (f(a)) is defined. (lim x rightarrow a f(x)) exists. (lim x rightarrow a f(x)=f(a)).

Solution Steps

To determine if \( f(2) \) is defined, we look at the value of the function at \( x = 2 \). From the graph, we see that there is a filled circle at \( (2, 3) \), indicating that \( f(2) = 3 \). Therefore, \( f(a) \) is defined at \( x = 2 \).

To determine if the limit exists, we need to check the left-hand limit and the right-hand limit as \( x \) approaches 2. From the graph:

• As \( x \) approaches 2 from the left, \( f(x) \) approaches 1.
• As \( x \) approaches 2 from the right, \( f(x) \) approaches 3.

Since the left-hand limit (\( 1 \)) and the right-hand limit (\( 3 \)) are not equal, \( \lim_{{x \to 2}} f(x) \) does not exist.

Final Answer