the graph below is the function f(x). determine which one of the following rules for continuity is first…

the graph below is the function f(x). determine which one of the following rules for continuity is first violated at a = - 1. of(a) is defined. olim f(x) exists. x→a olim f(x)=f(a). x→a

the graph below is the function f(x). determine which one of the following rules for continuity is first violated at a = - 1. of(a) is defined. olim f(x) exists. x→a olim f(x)=f(a). x→a

Answer

Explanation:

Step1: Check if (f(a)) is defined

At (a=-1), from the graph, there is a filled - in circle at (x = - 1) and (y=-1), so (f(-1)=-1) and (f(a)) is defined.

Step2: Check if (\lim_{x\rightarrow a}f(x)) exists

As (x) approaches (-1) from the left, the function values approach (-3). As (x) approaches (-1) from the right, the function values approach (-1). Since the left - hand limit (\lim_{x\rightarrow - 1^{-}}f(x)=-3) and the right - hand limit (\lim_{x\rightarrow - 1^{+}}f(x)=-1), (\lim_{x\rightarrow - 1}f(x)) does not exist. Since the limit does not exist, we don't need to check the third condition.

Answer:

(\lim_{x\rightarrow a}f(x)) exists.