the graph below is the function f(x)\ndetermine which one of the following explains why continuity is…

Brief Explanations:

A function (y = f(x)) is continuous at (x=a) if (\lim_{x\rightarrow a}f(x)=f(a)). Looking at the graph at (x = - 1), we can see that the function value (f(-1)) is defined (the solid - dot at (x=-1,y = - 2)). To check the limit as (x\rightarrow - 1), we consider the left - hand limit and the right - hand limit. The left - hand limit and the right - hand limit as (x\rightarrow - 1) approach the open - dot value at (y = 3), and (f(-1)=-2). So (\lim_{x\rightarrow - 1}f(x)) exists (the left - hand and right - hand limits are equal) and (f(-1)) exists, but they are not equal.

Answer:

(\lim_{x\rightarrow a}f(x)) and (f(a)) exist but are not equal.