Question

the graph below is the function f(x)
determine which one of the following explains why continuity is violated at x = - 1.
lim_{x→a} f(x) and f(a) exist but are not equal.
lim_{x→a} f(x) does not exist.
f(a) is undefined.

Explanation:

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

Answer:

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